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\titlefigurecaption{A pair of counterpropagating beams induce an instability that generates transverse optical patterns. (A) Two spots form the unperturbed far-field pattern. (B) A weak beam incident at an angle to the pump beam axis causes the generated pattern to rotate. From Ref.~\cite{Dawes_2005aa}}

%%% Give an abstract text:
\abstract{We review recent theoretical and experimental efforts
toward developing an all-optical switch based on transverse optical
patterns. Transverse optical patterns are formed when
counterpropagating laser beams interact with a nonlinear medium. A
perturbation, in the form of a weak switch beam injected into the
nonlinear medium, controls the orientation of the generated patterns.
Each state of the pattern orientation is associated with a state of
the switch. That is, information is stored in the orientation state. A
realization of this switch using a warm rubidium vapor shows that it
can be actuated by as few as 600~$\pm 40$~photons with a response
time of 5~$\mu$s. Models of nonlinear optical interactions in
semiconductor quantum wells and microresonators suggest these
systems are also suitable for use as fast all-optical switches using
this same conceptual design, albeit at higher switching powers.}

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\title{Transverse optical patterns for ultra-low-light-level all-optical switching}

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\author{Andrew M. C. Dawes,\inst{1,*} Daniel J. Gauthier,\inst{2}
Stefan Schumacher,\inst{3,4} N.H. Kwong,\inst{3} R. Binder\inst{3} and Arthur L. Smirl\inst{5}}

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\institute{%
  Department of Physics, Pacific University, Forest Grove, OR 97116, USA
\and
  Department of Physics, Fitzpatrick Institute for Photonics, and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, USA
  \and
  College of Optical Sciences,
             University of Arizona,
             Tucson, Arizona 85721, USA
             \and Physics Department, School of Engineering and Physical Sciences, Heriot-Watt University - Edinburgh, EH14 4AS, UK
             \and  Laboratory for Photonics and Quantum Electronics, 138
IATL, University of Iowa, Iowa City, Iowa 52242, USA
}

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\mail{e-mail: dawes@pacificu.edu}

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\titlerunning{Transverse patterns \& all-optical switching}
\authorrunning{A. M. C. Dawes, D. J. Gauthier, et al.}
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%%% Give some keywords and PACS code(s) for the article, if you have:
\keywords{all-optical, switching, nonlinear, transverse, counterpropagating, excitons, polaritons}
\pacs{42.65.Pc,42.79.Ta,42.65.Sf,71.35.-y,71.36.+c}


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\section{Introduction} % (fold)
\label{sec:introduction}

In the past decade, the rapid bandwidth increase in communication networks has
been enabled by advances in opto-electronic technology. However, bandwidth
improvements cannot continue indefinitely as opto-electronic devices face
thermal dissipation limits that are fundamental to processing information in
the electronic domain \cite{Cavin_2006aa}. Photonics offers a wide range of
novel information processing technologies with the potential for much greater
bandwidth. In particular, devices that process information in the optical
domain can operate on parallel channels, with high bandwidth, and with
markedly higher information density.

To process information all-optically, beams of light must interact with one
another, which can only occur in nonlinear media. Optical nonlinearities are
typically weak, requiring high intensities in order to generate significant
effects. A current goal in the field of nonlinear optics is to reduce the power required for nonlinear interactions. One application of nonlinear optics is the development of efficient telecommunications devices where strong nonlinearities reduce device power requirements. In addition, there is a need to operate devices at the ultimate, single-photon level. Few-photon or ultra-low-light nonlinear optics has many potential applications in quantum information science
\cite{Duan_2001aa}. Several recent experiments in the photonics field have
demonstrated ultra-low-light nonlinear optical effects using various
techniques including high-finesse cavities \cite{Hood_1998aa,Birnbaum_2005aa},
plasmonic nanostructures \cite{Chang_2007ac}, and quantum-interference effects
such as electromagnetically-induced transparency
\cite{Harris_1997aa,Harris_1998aa,Harris_1999aa,Zhang_2007aa}.

These approaches each achieve remarkable sensitivity; however, as we explain
below, they do not necessarily satisfy the requirements for use as a scalable
all-optical network element. Another approach, one that is the subject of this
review, is based on the control of transverse patterns generated by nonlinear
optical interactions. This pattern-based approach achieves a level of
sensitivity that is comparable to other methods in addition to satisfying the
requirements of scalability.

This review describes recent progress in the field of nonlinear optics that
has demonstrated all-optical switches that are capable of controlling one beam
of light with another. These devices exploit the inherent sensitivity of
pattern-forming instabilities to weak perturbations and are based on
transverse optical patterns that change orientation in the presence of a weak
control beam, or \emph{switch} beam. This article is designed as a review, but
also contains relevant results from our recent work. The review is arranged as
follows. The next section introduces the concepts of nonlinear pattern
formation and provides context for the application of pattern-forming systems
to problems in optical switching. Section \ref{sec:all_opt_switching}
describes various approaches to all-optical switching, and introduces common
metrics for comparing different devices. Section \ref{sec:experiment}
summarizes recent experimental results using an all-optical switch based on
transverse optical patterns that are formed in a counterpropagating-beam
system. The data presented in Sec.~\ref{sec:experiment} is new and shows a
marked improvement in comparison to the results that were reported previously
\cite{Dawes_2008aa}. The improvement was achieved by a careful optimization of
the experimental system. Section \ref{sec:numerical_results} presents new
numerical results obtained by simulating the interaction of Gaussian beams
counterpropagating through a medium that exhibits Kerr-like nonlinearity. In
Sec.~\ref{sec:semiconductor_systems}, we describe recent results based on
simulations of a related system where beams counterpropagate in a
semiconductor medium which exhibits excitonic nonlinearities. Finally, a
discussion of future directions is given in Sec.~\ref{sec:future_directions}.

% section introduction (end)

\section{Pattern Formation} % (fold)
\label{sec:patterns}

The emergence of regular structure from natural processes has been observed
throughout history. Found in nearly every field of science, patterns are one
of the most recognizable signatures of a nonlinear dynamical system. The
mathematical tools developed in the field of dynamics have been used
successfully to describe a wide range of pattern forming systems in biology,
chemistry, and computer science
\cite{Cross_1993aa,Strogatz_2001aa}.

The quantitative description of pattern formation requires a study of the
system dynamics and their stability relative to perturbations. For spatially
extended systems, the stability of the Fourier modes of the system are of
interest. Hence, if infinitesimal perturbations applied to a specific mode
grow as the system evolves, that mode can give rise to an instability.
Instabilities such as this are responsible for pattern formation in systems
with two or more dimensions.

The term \emph{pattern selection} refers to the tendency of the system to
exhibit patterns with a certain symmetry or orientation. Understanding the
pattern selection process is of fundamental importance to understanding the
patterns observed in the system. Many patterns are allowable solutions
to the dynamics equations of the system, yet only a subset of the allowed
patterns are typically exhibited. Patterns are selected both by constraints on
the system and by the dynamics of the system. The optical patterns that are
the subject of this review exhibit pattern selection by both mechanisms,
although primarily via the system dynamics, in particular through external
forcing \cite{Cross_1993aa}.

Given a specific system, and thus specific allowed solutions, control of the
generated patterns is limited to choosing from among these solutions. Hence,
it is through controlling pattern selection that one can control the pattern
generated by a system. Attempting to control the spontaneous patterns formed
by nonlinear processes is not an intrinsically new idea. In fact, attempts to
control many aspects of nature (i.e., weather, ocean currents, tides, and
wind) are simply attempts to control the patterns that arise from nature's
fundamental processes. There are, however, new applications for controllable
pattern-formation, and one such application---controlling the flow of optical
information---is described in the remainder of this review.

% section patterns (end)

\section{All-optical switching} % (fold)
\label{sec:all_opt_switching}

An all-optical switch is a device that allows the control of one beam of light
with another. Two fundamental properties of a switch are that the device
exhibit at least two distinguishable states, and that the device input and
output are distinguishable. There are many possible configurations where the
switch can change the output power, direction, or state of polarization of a
beam of light that is either propagating through a nonlinear medium or
generated within the medium.

One simple all-optical switch that has been demonstrated in a wide variety of
materials is based on the intensity-dependent refractive index of transparent
nonlinear optical media. The intensity-dependent refractive index leads to a
nonlinear phase shift experienced by a wave propagating through the medium.
This effect allows light-by-light control if such a medium is inserted in one
arm of an interferometer \cite{Boyd_2002aa}. The output state of the interferometer could then be controlled by changing the phase shift experienced by one beam, i.e., by changing the optical path length of one arm of the interferometer. The phase shift depends on the total optical intensity incident on the nonlinear material, so, if a strong control field is applied, and assuming the signal field is weak $I_\mathrm{signal}<<I_c$, the nonlinear phase shift is given by
\begin{equation}
	\phi_\mathrm{nl}=2\frac{\omega}{c}n_2 I_c L,
\end{equation} 
where $\omega$ is the angular frequency, $c$ is the speed of light in vacuum,
$L$ is the length of the medium, $n_2$ is the nonlinear index of refraction,
and $I_c$ is the intensity of the control field. For high-contrast switching,
the control beam must be of sufficient strength to cause a significant change
in the phase of the signal beam, $\phi_\mathrm{nl}\sim\pi$.

A second type of all-optical switch relies on the properties of a
\emph{saturable absorber}. The absorption experienced by a wave propagating
through a homogeneously-broadened medium that exhibits saturable absorption
depends on the intensity, and decreases for increasing intensity following the
relation \cite{Boyd_2002aa}
\begin{equation}
  \label{eqn:saturable_absorption}
    \alpha=\frac{\alpha_0}{1+I/I_\mathrm{sat}},
\end{equation}
where $\alpha_0$ is the absorption coefficient experienced by a weak field and
$I_\mathrm{sat}$ is the saturation intensity.

In order to realize a switch based on saturable absorption, a strong control
beam and a weak signal beam co-propagate through a material that exhibits
saturable absorption. The control beam, in this case, must be of sufficient
intensity to saturate the atomic response. Saturation of a two-level system
corresponds to moving a significant amount of the atomic population from the
ground state to the excited state. In order to maintain population in the
excited state, one photon must be incident on each atom per excited state
lifetime. Quantitatively, this condition is \cite{Siegman_1986aa}
\begin{equation}
  \label{eqn:saturation_intensity}
    I\approx I_\mathrm{sat}=\frac{\hbar \omega}{\sigma \tau_\mathrm{sp}},
\end{equation}
where $\tau_\mathrm{sp}$ is the excited state lifetime, $\hbar\omega$ is the
photon energy, and $\sigma$ is the atomic cross section. Equation
(\ref{eqn:saturation_intensity}) must be modified for the case of a material
undergoing optical pumping where the population is redistributed in time
$\tau_g$. Hence, to maintain saturation, one photon must be incident on each
atom per $\tau_g$, i.e., the relevant time scale is instead the ground-state
lifetime $\tau_g$. For a pair of isolated levels driven by resonant light,
without collisional or Doppler broadening, the cross section has a maximum
value of
\cite{Boyd_2002aa,Siegman_1986aa}
\begin{equation}
  \label{eqn:sigma}
  \sigma_\mathrm{max}=\frac{3\lambda^2}{2\pi},
\end{equation}
where $\lambda$ is the wavelength of incident light.

As an example, a cloud of cold-trapped rubidium atoms contained in a
magneto-optic trap (MOT) satisfies the requirements for maximizing $\sigma$.
The saturation intensity $I_\mathrm{sat}=3$~mW/cm$^2$ for
$\tau_\mathrm{sp}=25$~ns and $\lambda=780$~nm. Thus, for a beam with radius
2~mm, an optical power of 0.4~mW is required to actuate a
saturation-based switch using such a Rb-MOT medium. Of course, one limitation
of the saturation-based switch is that the switching beam must be weak
relative to the saturation intensity. In the MOT case described here, the
saturation intensity corresponds to a 2~mm diameter beam with a power of
$P_\mathrm{sat}=80$~$\mu$W, hence the power of the signal beam (the beam being
turned on or off) must be much lower,
$P_\mathrm{signal}<<P_\mathrm{sat}=80$~$\mu$W. Therefore, in the MOT example,
the maximum allowed signal $P_\mathrm{signal}<<80$~$\mu$W is significantly
weaker that the required switch beam power, which is equal to 0.4~mW.

To establish a convenient metric for comparing different all-optical switches
having different geometries, we quantify the energy density of the control
field in units of photons per $\lambda^2/(2\pi)$ \cite{Harris_1998aa}. In
principle, a larger device that operates at $n$ photons per $\lambda^2/(2\pi)$
can be scaled to have transverse dimension equal to the diffraction limit
($\lambda^2$) and operate with only $n$ photons. The relationship between the
saturation intensity and $\sigma_\mathrm{max}$ is such that an energy density
that corresponds to approximately one photon per \lambdasquared\ is sufficient
to saturate a two-level transition. Of course, energy of this density must be
applied for at least the lifetime of the excited state; otherwise, saturation
will not occur. The assumption that saturation of the atomic transition is
required for observing high-contrast all-optical switching led to the early
conclusion that all-optical switches must operate with at least one photon per
\lambdasquared ~\cite{Keyes_1970aa}. As this review shows, many recent
all-optical switching schemes beat this limit by several orders of magnitude
through various approaches. However, the metric remains useful as a tool for
comparison across designs.

Sensitivity, measured in photons per $\lambda^2/(2\pi)$, is only one
measure of switch performance. Depending on the design of a given
all-optical switch, there are a wide range of applications each of
which has additional requirements. The next section reviews two
general application classes and outlines the requirements a
switching  device must satisfy for practical use.

% section all_optical_switching_via_saturation_of_a_two_level_atom (end)

\subsection{Applications} % (fold)
\label{sec:criteria}

Switches can be used in two classes of applications: information networks and
computing systems. In each of these applications, information can be stored in
either classical or quantum degrees of freedom. Hence, the requirements for a
device vary depending on the intended application.

Classical, all-optical networks require switches to reliably redirect or gate
a signal depending on the presence of a control field at the device input.
Ideally, the switch shows large contrast between on and off output levels and
can be actuated by low input powers. If the network carries quantum
information, the switch must be triggered by an input field containing only a
single quanta (photon). Additionally, in the quantum case, the quantum state
of the transmitted signal field must be preserved.

If a switch is to be used as a logic element in a classical computing system,
it must have the following characteristics: input-output isolation,
cascadability, and signal level restoration \cite{Keyes_2006aa}. Input-output
isolation prohibits the device output from having back-action on the device
input. Cascadability requires that a device output have sufficient power to
drive the input of at least two identical devices. Signal level restoration
occurs in any device that outputs a standard signal level in response to a
wide range of input levels. That is, variations in the input level do not
cause variations in the output level. Switching devices that satisfy these
requirements are considered scalable devices; the properties of
the individual device are suitable for scaling from one device to a network
of many devices.

While scalability describes important properties of a switching device,
sensitivity provides one way to quantify its performance. A highly sensitive
all-optical switch can be actuated by a very weak optical field. Typical
metrics for quantifying sensitivity are: the input switching energy (in
Joules), the input switching energy density (in photons per
$\sigma=\lambda^2/2\pi$) \cite{Harris_1998aa,Keyes_1970aa}, and the total
number of photons in the input switching pulse.

One may not expect a single device to satisfy all of the requirements for
these different applications. For example, a switch operating as a logic
element should output a standard level that is insensitive to input
fluctuations. This may be at odds with quantum-switch operation where the
device must preserve the quantum state of the signal field. An interesting
question arises from these requirements: What happens when a classical switch
is made sensitive enough to respond to a single photon? Reaching the level of
single-photon sensitivity has been the goal of a large body of recent work
that is reviewed below.

% section overview (end)

\subsection{Previous Research on Low-Light-Level Switching} % (fold)
\label{sec:previous_switches}

Two primary approaches to low-light-level switching have emerged, both of
which seek to increase the strength of the nonlinear coupling between light
and matter. The first method uses fields and atoms confined within and
strongly coupled to a high-finesse optical cavity. The second method uses
traveling waves that induce quantum interference within an optical medium and
greatly enhance the effects of light on matter. These methods have been
recently reviewed in Ref.~\cite{Dawes_2008aa}, the following brief discussion
and Table~\ref{table:schemes} summarize the key differences between various
methods.

Cavity quantum-electrodynamic (CQED) systems offer very high sensitivity by
decreasing the number of photons required to saturate the response of an atom
that is strongly coupled to a mode of the cavity. Strongly-coupled CQED
systems show a nonlinear optical response to fields corresponding to much less
than a single cavity photon \cite{Hood_1998aa}, and have also demonstrated the
photon blockade effect where the arrival and absorption of one photon prevents
subsequent absorption of a second photon \cite{Birnbaum_2005aa}.

A different technique for all-optical switching in cavities relies on creating
and controlling cavity solitons in vertical cavity surface emitting lasers
(VCSELs) \cite{Hachair_2005aa}. A VCSEL can be prepared for cavity solitons by
injecting a wide holding beam along the cavity axis. A narrow ``write'' beam
superimposed on the larger holding beam induces a cavity soliton that persists
typically until the original holding beam is turned off. This system naturally
serves as a pixel-based optical memory, where solitons are written to and
stored in the cavity field.

\begin{table*}
  \begin{center}
  \begin{tabular}{l c c c c c}
    \hline
    Switch design&Reference&Switching photons&photons$/\sigma$&$\tau_r$&Cascadable\\ \hline
    CQED&Hood \etal\cite{Hood_1998aa}&1&$10^{-4}$&25 ns&N\\
    CQED&Birnbaum \etal\cite{Birnbaum_2005aa}&1&$10^{-5}$&25 ns&N\\
    EIT&Zhang \etal\cite{Zhang_2007aa}&20&$10^{-5}$&0.7~$\mu$s&N\\
    Parametric Down Conversion&Resch \etal\cite{Resch_2002aa}&1&-&-&N\\
    6-wave mixing&Kang \etal\cite{Kang_2004aa}&$10^8$&2&0.54~$\mu$s&N\\
    MI in fiber&Islam \etal\cite{Islam_1988aa}&$\sim$2000&24&50~ps&Y\\
    Plasmonic nanowire*&Chang \etal\cite{Chang_2007ac}&1&-&-&Y\\ 
    3-wave mixing OPA&Han \etal\cite{Han_2008aa}&0.75&$10^{-5}$&400~fs&Y\\
    Fiber OPA&Andrekson \etal\cite{Andrekson_2008aa}&150&-&5~ps&Y\\
\hline\hline
    Pattern-based switches&&&&&\\ \hline
    VCSEL solitons&Hachair \etal\cite{Hachair_2005aa}&24,000&140&500 ps&-\\
    semiconductor quantum wells*&Kheradmand \etal\cite{Kheradmand_2008aa}&-&-&$<$100 ns&Y\\
    semiconductor quantum wells*&Schumacher \etal\cite{Schumacher2007e}&-&-&-&Y\\
    Rubidium vapor&Dawes \etal\cite{Dawes_2008aa}&600&$10^{-3}$&3~$\mu$s&Y\\ \hline
  \end{tabular}
  \end{center}
  \caption{Comparison of all-optical switching schemes. *Results of numerical simulation.}
  \label{table:schemes}
\end{table*}

In contrast to cavity systems, traveling wave approaches can operate with
multi-mode optical fields and also achieve few-photon sensitivity. Recent
progress in traveling-wave low-light-level nonlinear optics has been made
through the techniques of electro-magnetically induced transparency (EIT)
\cite{Harris_1997aa,Schmidt_1996aa,Zibrov_1999aa,Braje_2003aa,Chen_2005aa,Zhang_2007aa}.
As an example, Harris and Yamamoto \cite{Harris_1998aa} proposed a switching
scheme using the strong nonlinearities that exist in specific states of
four-level atoms where, in the ideal limit, a single photon at one frequency
causes the absorption of light at another frequency. To achieve the lowest
switching energies, the narrowest possible atomic resonances are required,
which can be obtained in complex experimental environments such as trapped
cold atoms
\cite{Harris_1999aa,Yan_2001aa,Braje_2003aa,Chen_2005aa,Zhang_2007aa}.

Other low-light-level all-op\-ti\-cal switch\-ing experiments have also been
demonstrated recently in traveling-wave systems. By modifying the correlation
between down-converted photons, Resch \etal \cite{Resch_2002aa} created a
conditional-phase switch that operates at the single photon level. Using
six-wave mixing in cold atoms, Kang \etal \cite{Kang_2004aa} demonstrated
optical control of one field by applying another input switching field.

Another approach combines the field enhancement offered by optical cavities
with the strong coupling of coherently prepared atoms. Bistability in the
output of a cavity filled with an EIT medium that also shows large Kerr-type nonlinearity \cite{Wang_2002aa}
exhibits switching. Photonic crystal nanocavities
have also shown bistability switching \cite{Tanabe_2005aa}.
Taking a different approach, Islam \etal \cite{Islam_1988aa} exploit a
modulational instability in an optical fiber interferometer to gate the
transmission of a strong beam by injecting a weak beam.

Many of these other systems satisfy some, but not all, of the criteria for
scalability. Of the systems just discussed, CQED systems are designed to
operate in a single field mode, which limits the number of input and output
channels to one per polarization. Additionally, all fields are strongly
coupled to the atom-cavity system so the control and signal fields must be of
comparable strength. Thus a CQED switch is not cascadable. EIT systems suffer
from a similar drawback in that the input and output fields are required to
have the same power, making them not cascadable. Another highly sensitive
system, the modulational-instability fiber interferometer, is both cascadable
and exhibits signal level restoration. In several ways, the latter system is
similar to pattern-based devices: it exploits the sensitivity of instabilities
and uses a sensitive detector (an interferometer in their case and pattern
orientation in the present case) to distinguish states of the switch.

Recently, there has been a proposal that does not use cavities or
traveling optical fields, but instead takes advantage of photon-induced
surface plasmons excited in a conducting nano-wire that couple strongly to a
two-level emitter placed nearby. This strong coupling enables effects that are
similar to those observed in CQED. Specifically, Chang \etal
\cite{Chang_2007ac} suggest that a system consisting of a nano-wire coupled to
a dielectric waveguide can be used to create an optical transistor that is
sensitive to a single photon. Photons in the dielectric waveguide are
efficiently coupled to plasmons that propagate along the nanowire. A two-level
emitter placed close to the nanowire has a strong effect on the plasmon
transmission. The absorption of a single photon by the emitter is sufficient
to change the nanowire from complete plasmon reflection to complete plasmon
transmission. If implemented as proposed, a surface-plasmon transistor could
operate with single-photon input levels, and gate signals containing many
photons.

Finally, another general approach to low-light all-optical switching has been demonstrated based on optical parametric amplification (OPA). Two specific results highlight this technique which shows promise for ultra-fast all-optical switching. First, using three-wave mixing OPA in a beta-barium borate crystal, Han \etal \cite{Han_2008aa} demonstrate a cascadable all-optical switch that can be actuated by a field with an average of 0.75 photons. This switch operates in the ultrafast regime and can be actuated in 400~fs. However, the contrast ratio between the on and off states, 1.032:1, is significantly smaller than in other switching schemes. In this proof-of-principle demonstration, measuring the switch response required phase-sensitive detection. In a fiber-optic system, Andrekson \etal \cite{Andrekson_2008aa} operate the OPA in the saturated regime and observe high-contrast ($>$3~dB) switching with as few as 150 photons. This fiber-based switch is cascadable and has a response time on the order of 5~ps. Specific advantages of this system are that it is fiber-based and operates at telecommunications wavelengths. One drawback, however is that the ratio between the strong signal power and the weak control power is fixed by the OPA gain. This limitation can be relaxed by using a longer fiber or fiber with larger nonlinearity. As demonstrated, 500~m of highly nonlinear fiber requires strong signal powers on the order of one watt.

Many all-optical switches have been successfully demonstrated over a period
spanning several decades. However, in almost every case, one or more important
features is missing from the switching device. With the requirements of
scalability and sensitivity in mind, this review presents recent results
generated from a new approach to all-optical switching.

% subsection previous research (end)

\subsection{Switching with Transverse Optical Patterns} % (fold)

A new approach to all-optical switching is to exploit collective instabilities
that occur when laser beams interact with a nonlinear medium
\cite{Dawes_2005aa}. One such collective instability occurs when laser beams
counterpropagate through an atomic vapor. In this configuration, it is known
that mirror-less parametric self-oscillation gives rise to stationary,
periodic, or chaotic behavior of the intensity
\cite{Silberberg_1984aa,Khitrova_1988aa} and/or polarization
\cite{Gaeta_1987aa,Gauthier_1988aa,Gauthier_1990aa}.

Another feature of counterpropagating beam instabilities is the formation of
transverse optical patterns, i.e., the formation of spatial structure of the
electromagnetic field in the plane perpendicular to the propagation direction
\cite{Petrossian_1992aa,Lugiato_1994aa}. This is also true for recent
experiments where a wide variety of patterns can be generated, including rings
and multi-spot off-axis patterns in agreement with previous experiments
\cite{Petrossian_1992aa,Grynberg_1988aa,Gauthier_1990aa}.

Building an all-optical switch from transverse optical patterns combines
several well-known features of nonlinear optics in a novel way. Near-resonance
enhancement of the atom-photon coupling makes our system sensitive to weak
optical fields. Using optical fields with a counterpropagating beam geometry
allows for interactions with atoms in specific velocity groups leading to
sub-Doppler nonlinear optics without requiring cold atoms. Finally, using the
different orientations of a transverse pattern as distinct states of a switch
allows one to maximize the sensitivity of the pattern forming instability.
Instabilities, by nature, are sensitive to perturbations, so by combining
instabilities with resonantly-enhanced, sub-Doppler nonlinearities,
researchers in this field have created a switch with very high sensitivity.

% subsection switching with patterns (end)

% section all_opt_switching (end)

\section{Switching in warm Rb vapor} % (fold)
\label{sec:experiment}

A pair of beams counterpropagating through a nonlinear optical medium give
rise to patterns formed by light that is spontaneously emitted at an angle to
the pump-beam axis. This section presents experimental results of pattern
formation in a counterpropagating beam system where a sample of warm rubidium
vapor serves as the nonlinear medium.

\subsection{Experimental apparatus} % (fold)
\label{sub:experimental_apparatus}

\begin{figure}[htbp]
  \begin{center}
    \includegraphics{Figure_1.eps}
  \end{center}
  \caption[System for observing pattern formation in rubidium vapor.]{Experimental setup for transverse optical pattern generation. The output of a frequency-stabilized cw Ti:Sapphire laser serves as the source. A polarizing beamsplitter (PBS1) separates the forward (cw) and backward (ccw) beams within the triangular ring cavity. The backward beam is brought into horizontal polarization by a half-wave plate ($\lambda$/2). The forward and backward beams counterpropagate through a warm $^{87}$Rb vapor contained in a 5-cm-long glass cell. A polarizing beam-splitter (PBS2) reflects instability-generated light in the vertical polarization which is observed by a CCD camera and avalanche photodiode (APD).}
  \label{fig:setup}
\end{figure}

A diagram of the atomic-vapor experimental setup is shown in
Fig.~\ref{fig:setup} \cite{Dawes_2008aa}. Two beams of light from a common
laser source counterpropagate through warm rubidium vapor contained in a glass
cell. The light source is a frequency-stabilized cw Ti:Sapphire laser, the
output of which is spatially filtered using a single-mode optical fiber with
an angled entrance face and a flat-polished exit face. The beam is then
collimated using a pair of convex lenses arranged as a telescope. The spot
size (1/e field radius), denoted by $w_0$, is controlled by the configuration
of the telescope, and the beam waist is located in the center of the vapor
cell. The power ratio between the pump beams is controlled by a half-wave
plate at the input of the first polarizing beam splitter (PBS1). We denote the
beam passing through PBS1 as the forward beam and the reflected beam as the
backward beam. A second half-wave plate in the backward beam path rotates the
polarization such that the pump beams are linearly polarized with parallel
polarizations.

The cell is fixed with length $L=5$~cm, and a diameter of 2~cm. The cell
contains a droplet of rubidium, melting point 39.3~$^{\circ}$C, which is in
equilibrium with rubidium vapor. The rubidium contained in the cell has not
been isotopically enriched and thus contains the two naturally abundant
isotopes: $\sim$ 72\% $^{85}$Rb, 28\% $^{87}$Rb. The cell is heated to
80~$^{\circ}$C corresponding to an atomic number density for $^{87}$Rb of
2$\times10^{11}$~atoms$/\mathrm{cm}^3$. The cell has uncoated quartz windows
that have fixed and opposing tilt angles of $\pm$11 degrees with respect to
the incident laser beams to prevent possible oscillation between the windows.
The cell has no paraffin coating on the interior walls that would prevent
depolarization of the ground-state coherence, nor does it contain a buffer gas
that would slow diffusion of atoms out of the pump laser beams. The
Doppler-broadened linewidth of the transition at this temperature is $\sim$550
MHz. To prevent the occurrence of magnetically-induced instabilities and
reduce Faraday rotation, a cylindrical $\mu$-metal shield surrounds the cell
and attenuates the ambient magnetic fields by a factor of $>\!10^3$. In order to
attenuate the static magnetic field created by the heaters coils, they are
placed outside the shielding.

A polarizing beam splitter (PBS2) placed in the beam path separates light
polarized orthogonally to the pump beam. This light, henceforth referred to as
\emph{output} light, is subsequently split with a 50/50 beamsplitter and then
observed simultaneously using any two of the following: a CCD-camera (Marshall
V-1050A), an avalanche photodiode (Hamamatsu C5460), or a photomultiplier tube
(Hamamatsu H6780-20) as shown in Fig.~\ref{fig:setup}.

% subsubsection experimental_apparatus (end)

\subsection{Instability generated light} % (fold)
\label{sub:instability_generated_light}

In an experimental setup similar to that described above, Dawes \etal
\cite{Dawes_2005aa,Dawes_2008aa} observe instability generated light (output
light) in the state of polarization orthogonal to that of the pump beams and
with the same frequency as the pump beams. The following sections describe
features of the instability as well as the conditions required for observing
pattern formation. The experimental variables are the frequency of the pump
light, the alignment and intensity of the pump beams, and the pump beam waist
$w_0$.

The fixed cell length has been chosen to balance large optical depth, which
increases with increasing $L$, and available transverse modes, which decrease
with increasing $L$. The Fresnel number
\begin{equation}
  \label{eqn:fresnel_num}
  \mathcal{F}=\frac{w_0^2}{\lambda L},
\end{equation}
quantifies the number of transverse modes supported by the geometry where
$\lambda$ is the wavelength \cite{Siegman_1986aa}. Dawes \etal have observed
light generated off-axis for Fresnel numbers between 1.9 and 7.8,
corresponding to $w_0$ between 270~$\mu$m and 550~$\mu$m, respectively. The
results reviewed here correspond to $w_0=455$~$\mu$m with $\lambda=780$~nm, or
$\mathcal{F}=5.3$.

The other fixed parameter, the temperature, has been chosen based on
optimizing the pattern-formation. Changing the temperature of the cell affects
both the temperature of the atomic vapor and the atomic number density. For
this work, changing the atomic number density primarily affects the optical
depth of the vapor. By varying the cell temperature, and observing the amount
of optical power generated by the instability, Dawes \etal found that the
optimum temperature is $80^\circ$~C. Fitting the absorption profile at
$T=80^\circ$~C to a model for Rubidium absorption, they find that the maximum
Doppler-broadened optical depth at this temperature is $\alpha L\simeq55$
\cite{Dawes_2008aa}.

\subsubsection{Pump-beam frequency} % (fold)
\label{ssub:pump_beam_frequency}

The power of the output light is maximized (and the threshold for
the instability is lowest) when the frequency of the pump beams is set near an
atomic resonance, i.e., the instability occurs near either the D$_1$ or
D$_2$ transition of $^{87}$Rb. The results reviewed here are for pump-beam
frequencies near the D$_2$ transition ($^5\text{S}_{1/2} \rightarrow
{^5\text{P}_{3/2}}$, 780~nm wavelength).

Figure~\ref{fig:scan} shows the power of the output light as a function of
pump frequency detuning, defined as $\Delta=\nu-\nu_{F=1,F'=1}$ in cycles/s.
One can observe several sub-Doppler features, where the maximum power emitted
in the orthogonal polarization occurs when the laser frequency $\nu$ is tuned
$\Delta=+25$~MHz. The Doppler-broadened linewidth of the transition at this
temperature is $\sim 550$~MHz, hence, the generated light is only emitted for
pump frequencies in a narrow range within the Doppler profile. For this
detuning, 3.5~$\mu$W of output light is generated in the forward direction,
indicating that $\sim 1$\% of the incident pump power is being converted to
the orthogonal polarization. Because the detuning is small relative to the
Doppler width, a significant amount of the pump light is absorbed by the
medium. Although the medium is optically thick ($\alpha L\sim55$), there is
substantial bleaching with 415~$\mu$W of forward pump-beam power, which allows
transmission of 50~$\mu$W of forward pump light. Of this transmitted power,
3.5~$\mu$W, or $\sim 7$\%, is converted to the orthogonal polarization
\cite{Dawes_2008aa}. In the next section, we discuss how the presence of
absorption affects the instability.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics{Figure_2.eps}
  \end{center}
  \caption[Generated power as a function of pump
  detuning]{Instability-generated optical power as a function of pump laser
  frequency detuning ($\Delta = \nu - \nu_{F=1,F'=1}$). The plot shows the
  power generated in the forward direction and in the state of polarization
  orthogonal to that of the pump beams. These data correspond to a single scan
  through the $^5\text{S}_{1/2}(F=1) \rightarrow {^5\text{P}_{3/2}}(F')$
  transition in $^{87}$Rb from low to high frequency. The bold tick marks at
  the top of the frame indicate the hyperfine transitions labeled by FF',
  where F (F') is the ground (excited) state quantum number. Pump beam power
  levels for this data are 415 $\mu$W (forward) and 145 $\mu$W (backward), and
  $w_0=455\,\mu$m.}
  \label{fig:scan}
\end{figure}

The instability clearly occurs on the blue-detuned (high-frequency) side of
the $^5\text{S}_{1/2}(F=1) \rightarrow {^5\text{P}_{3/2}}(F'=1)$ transition.
This is the side of the resonance where the nonlinear refractive index has a
positive value and hence self-focusing is expected to occur. This experimental
observation agrees with theoretical models and can be explained using a simple
argument based on weak-wave retardation \cite{Chiao_1966aa}. The forward
four-wave-mixing process can only become phase matched for off-axis beams if
the nonlinear refractive index $n_2$ has a positive value, i.e., on the
high-frequency side of an atomic resonance \cite{Boyd_2002aa}. If $n_2$ is
negative, the off-axis wavevectors are shortened, and thus cannot be
phase-matched to the pump-beams regardless of the angle $\theta$
\cite{Grynberg_1988ab,Boyd_2002aa}.

The width of the feature shown in Fig.~\ref{fig:scan} changes with pump power
such that it is narrower near threshold. This change indicates that phase
matching depends on the pump power such that a wider range of frequencies are
phase-matched for larger pump powers. The amount of power generated in the
orthogonal polarization is also lower near threshold and increases linearly
with increasing pump power as described in the following section.

% subsubsection pump_beam_frequency (end)

\subsubsection{Pump-beam intensity} % (fold)
\label{ssub:pump_beam_intensity}

The instability observed in this system has a very low threshold; the power
required to induce self-oscillation is less than 1~mW, which is comparable to
the results obtained from coherently-prepared atomic media
\cite{Zibrov_1999aa}. A common way to measure the instability threshold for a
setup with counterpropagating beams is to fix the power of one of the beams
and measure the output power as a function of the power in the second pump
beam \cite{Gauthier_1988aa,Zibrov_1999aa}. For a pump-beam detuning of
$\Delta=+25$~MHz and with a fixed forward pump power of 415~$\mu$W, Dawes
\etal find that the backward pump power threshold is $\sim$75~$\mu$W,
corresponding to a total pump power of 490~$\mu$W \cite{Dawes_2008aa}.

Another way to measure the instability threshold is to determine the minimum
total pump power necessary to generate output light. We find that there is an
optimum ratio of forward power to backward power of $\sim$3-to-1. At this
ratio, the threshold for off-axis emission is 385~$\mu$W, which is slightly
lower than the threshold measured with fixed forward beam power. Dawes \etal
report patterns and switching with $\sim$560~$\mu$W of total power,
corresponding to 40\% above threshold. Both threshold measures demonstrate
that the nonlinear process that generates new light is induced by a pair of
very weak fields. This indicates strong nonlinear matter-light interaction
comparable with the best reported results to date for warm-vapor
counterpropagating beam systems \cite{Zibrov_1999aa}.

For most of the early observations of nearly-degenerate instabilities, strong
pump fields were used (typically hundreds of mW)
\cite{Grynberg_1988aa,Petrossian_1992aa,Maitre_1995aa}. A considerably higher
threshold was reported for the first observation of polarization instabilities
in a sodium vapor \cite{Gauthier_1988aa}, where a threshold of tens of mW was
found when the pump fields were tuned near an atomic resonance. More recently,
Zibrov \etal \cite{Zibrov_1999aa} observed parametric self-oscillation with
pump powers in the $\mu$W regime using a more involved experimental setup
(``double-$\Lambda$'' configuration) designed specifically to lower the
instability threshold. In their experiment, atomic coherence effects increase
the nonlinear coupling efficiency. They report oscillation with several mWs of
total pump power. With 5~mW of forward-beam power, their instability threshold
corresponds to 20~$\mu$W in the backward beam. In contrast, the results
reviewed here demonstrate that spontaneous parametric oscillations are induced
by $\mu$W-power counterpropagating pump-beams without the need for special
coherent preparation of the medium. Furthermore, Zibrov \etal observed only
on-axis emission, whereas Dawes \etal found that off-axis emission requires
roughly half as much pump power as on-axis emission with our pump beam
configuration. In situations where low power and high sensitivity are
important, such as in all-optical switching, the lower instability threshold
may make off-axis instabilities preferable.

% subsubsection pump_beam_intensity (end)

\subsubsection{Patterns} % (fold)
\label{ssub:patterns}

In the context of all-optical switching, pattern formation is the most notable
feature of the counterpropagating beam instability described above. When the
pump beams are above threshold, i.e., have total power greater than
420~$\mu$W, generated light is emitted at an angle $\theta\simeq4$~mrad with
respect to the pump beam axis, as shown in Fig.~\ref{fig:cones}(a). A
perfectly symmetric system may at first be expected to generate light with
intensity that is distributed evenly around the azimuthal angle, and hence
would form a ring pattern in the far field. However, there are two effects
that prevent this ring pattern from being a stable steady-state of the present
system. First, symmetric nonlinear systems are known to spontaneously select
states that are in a symmetry group that is a subset of the original one. As
an example of this, simulations of counterpropagating beam systems exhibit
multispot patterns even when the pump beams are perfectly symmetric
\cite{Geddes_1994aa}. Second, perfect symmetry is unattainable in the
laboratory where imperfections in optical elements impart small perturbations
on the phase and amplitude of the beams. The instability responds to such
perturbations by generating patterns that are not cylindrically symmetric. For
this reason, the most common patterns reported consist of two, four, or six
spots in a variety of arrangements. In all cases, the spots are located along
the ring projected by the cones onto the detection plane as illustrated in
Fig.~\ref{fig:cones}(b-d).

\begin{figure}[htbp]
  \begin{center}
    \includegraphics{Figure_3.eps}
  \end{center}
  \caption{Light is generated along cones (blue) centered on the pump-beam
  axis when pump beams (red) of a sufficient intensity counterpropagate
  through warm rubidium vapor. A far-field detection plane shows patterns
  formed by the generated light. b) A ring pattern is expected for a perfectly
  symmetric system. c) Six spots form a hexagon, the typical pattern for pump
  beam powers more than 20\% above threshold. d) Two spots are observed just
  above threshold or when the pump beams are mis-aligned. From
  Ref.~\cite{Dawes_2005aa}}
  \label{fig:cones}
\end{figure}

Most theoretical treatments consider only the case where the pump beams are
strictly counterpropagating, corresponding to pump beams with equal and
opposite wavevectors. In experiments, it is common to have slight
mis-alignment, either intentional or accidental, between the pump beams.

With misaligned pump beams, the generated patterns change. This change is due
to a change in the phase-matching conditions for the different azimuthal
angles. Hence, there are different amounts of gain for different off-axis
beams, and for those each beam that experiences sufficient gain for
self-oscillation, a spot will be generated in the pattern. Alignment of the
pump beams provides one method for pattern selection in the
counterpropagating-beam system.

% subsection patterns (end)

\subsubsection{Secondary instability} % (fold)
\label{ssub:modulational_instability}

In addition to the instability responsible for pattern formation, the system
exhibits a secondary modulational instability (MI) that is manifested as
oscillations in the intensity of the generated light. The frequency of the
intensity oscillations due to this instability depends on the alignment of the
pump beams such that larger misalignment increases the oscillation frequency. For well-aligned beams, counterpropagating along a common axis,
the MI is generally suppressed as long as the pump-beam intensity is not
significantly far above threshold. Figure~\ref{fig:mionset} illustrates the
onset of this secondary instability for the case of slightly misaligned pump
beams. The threshold behavior described previously is evident here as well:
the power generated in the orthogonal polarization increases linearly above
385~$\mu$W total pump power. Also visible is the saturation of the
pattern-forming instability near 800~$\mu$W, where increasing the total pump
power no longer increases the generated power. The height of the vertical bars
indicates the peak-to-peak amplitude of oscillations due to the secondary MI.
There is a notable increase in the amplitude of the MI oscillations above
560~$\mu$W total pump power (indicated in the figure), and a significant
increase above 800~$\mu$W total pump power.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics{Figure_4.eps}
  \end{center}
  \caption[The onset of a secondary instability.]{The peak-to-peak
  oscillations generated by the secondary modulational instability are
  indicated by the vertical bars. The output light generated in the orthogonal
  polarization is plotted as a function of total pump-beam power. Data are
  collected with fixed forward-to-backward pump-beam power ratio of 3:1, and
  detuning $\Delta=+25$~MHz.}
  \label{fig:mionset}
\end{figure}

When the pump beams are made to counterpropagate with a small angle between
their axes, observing the photodetector signal with a spectrum analyzer
reveals a harmonic series with a fundamental frequency that increases for
larger angular separation of the pump beam axes. The spectrum for pump beams
misaligned with $\sim0.4$~mrad between the beam axes has a fundamental
frequency of 250~kHz \cite{Dawes_2008ab}. The atomic vapor system operating
with this amount of pump beam misalignment exhibits sensitive switching as
discussed in the next section

% subsubsection modulational_instability (end)

% subsection instability generated light (end)

\subsection{Switch response} % (fold)
\label{sub:switch_response}

To quantify the dynamic behavior of their switch, Dawes \etal inject a series
of pulses by turning the switch beam on and off with the EOM
\cite{Dawes_2008aa}. Spatially filtering the output pattern enables direct
measurement of the switch behavior. High-contrast switching is confirmed by
simultaneously measuring two output ports. Figure~\ref{fig:datafull}(a)
indicates the power of the injected switch beam as a function of time. The
signal from the off-state detector is shown in Fig.~\ref{fig:datafull}(b) and
is high when the switch beam is not applied and low during a switch-beam
pulse. The on-state detector is shown in Fig.~\ref{fig:datafull}(c) and shows
the opposite behavior: it is low when the switch beam is not applied and high
during each switch-beam pulse. These alternating signals demonstrate switching
of the power from one switch state to another with high contrast. Because the
switch reverts to the off state after the switch-beam is turned off, this
device acts as a non-latching switch. The total power generated in the pattern
is $\sim3\,\mu$W. Each aperture selects one of the two generated spots, so the
switch output power is $\sim1.5\,\mu$W per aperture. Of course, two apertures
could be used per switch state to transmit the full $\sim3\,\mu$W output.

One notable feature of the system response is the transition from complete
switching to partial switching. The first three pulses in
Fig.~\ref{fig:datafull} show that the on-state detector is fully illuminated
and the off-state detector is dark. This indicates that the switch beam has
caused complete rotation of the pattern and transferred all of the power from
the off-state spots to the on-state spots. For the last seven pulses in the
series, the system exhibits partial switching, where the on-state detector is
partially illuminated and the off-state detector is partially darkened. This
partial response indicates that the off-state spots are suppressed but not
extinguished when the switch beam is applied with less than 900~pW. Similarly,
the on-state spots are generated but not at full power. In this intermediate
regime, from 900~pW to $<$300~pW, the response depends on the input power.


\begin{figure}[htbp]
  \centering
    \includegraphics{Figure_5.eps}
  \caption{The switch responds to a series of ten pulses by transferring power
  from the \emph{off} state spots to the \emph{on} state spots. a) The switch
  beam power steadily decreases in power from 1.2~nW to 200~pW. b) The off
  spot is extinguished in the presence of the switch beam. c) The on spot
  power increases in the presence of the switch beam. The data shown are
  collected in a single shot that contains 22 additional ten-pulse sets with
  similar response. No signal averaging has been performed on the switch
  response data (b,c). The measured switch-beam power shown in (a) is averaged
  over 10 shots.}
  \label{fig:datafull}
\end{figure}

Barely visible in Fig.~\ref{fig:datafull} is a the secondary modulational
instability that causes small oscillations in the total output power. The
modulation period of this secondary instability (4~$\mu$s$=1/250$~kHz) and the
characteristic response time of the switch both correspond roughly to the
transverse transit time of a thermal atom through the pump beams. This is a
typical time scale for nonlinearities due to optical pumping.



\subsubsection{Switching photon number} % (fold)
\label{ssub:switching_photon_number}

To quantify the sensitivity of the system, Dawes \etal measure the response
time and calculate the number of photons $N_p$ required to actuate the switch.
The response time of the device $\tau_r$ is defined as the time between the
initial rising edge of the electronic signal driving the EOM and the point
where the on-spot signal crosses a threshold level set to roughly correspond
to a signal-to-noise ratio of $\sim$3~dB.\footnote{The SNR$\sim$3~dB criterion
corresponds to the threshold where the bit-error-rate decreases below 0.05,
\emph{i.e.}, it is the point where pulses can be correctly detected with
probability greater than 5\% \cite{Kish_2000aa}.} Results using this threshold
are shown in Fig.~\ref{fig:responsetime}(a) and we find that the measured
response time increases as the input switch beam power decreases.

\begin{figure}[htbp]
  \centering
    \includegraphics{Figure_6.eps}
  \caption{The response time $\tau_r$ and number of switching photons $N_p$ as
  a function of input power. Data are generated from 22 sequential traces like
  the one shown in Fig.~\ref{fig:datafull} acquired after a single trigger.
  The error bars indicate one standard deviation of the measured values. The
  solid line indicates the fit: $N_p = 7081 P_s + 404$ for $P_s$ in nW. It
  should be noted that the response times for the switch are on the order of a
  few $\mu$s, whereas the response time of the measurement system is $<
  35$~ns.}
  \label{fig:responsetime}
\end{figure}

The number of photons required to actuate the switch is given by $N_p=\tau_r
P_s/E_p$ where $\tau_r$ is the response time, $P_s$ is the switch beam power
and $E_p=2.54\times10^{-19}$~J is the photon energy. For ten switch-beam
powers between 510~pW and 35~pW, the response time is plotted in
Fig.~\ref{fig:responsetime}(a), and the number of switching photons is plotted
in Fig.~\ref{fig:responsetime}(b). The response time is longer for weak
switch-beam powers so the photon number decreases gradually as the input power
decreases. The data points indicate the average of 22 data points for each of
the ten switch-beam pulses, and the error bars represent one standard
deviation in the response times observed for each pulse.

The implication of the linear regression shown in
Fig.~\ref{fig:responsetime}(b) is that, in the limit as $P_s\rightarrow0$, the
number of switching photons $N_p\rightarrow400$. This would indicate that the
minimum number of photons capable of actuating the switch is above 400. The
final data point shown correspond to switching with $N_p=600\,\pm 40$, only
200 photons above this limit, and a factor of $\sim$5 lower than the first
reported observation of pattern-based all-optical switching with 2,700 photons
\cite{Dawes_2005aa}.\footnote{The error reported in this value of $N_p$ is a
combination of statistical error in the measurement of the switch-beam
power ($\sim$0.5\%), and statistical variations in the response time measured
for 22 sequential shots ($\sim$5\%).} It is also important to consider a possible threshold for the switch beam power ($P_s$). There must ultimately be a minimum switch beam power and a rough estimate of that threshold is simply the switch beam power that corresponds to 400 photons incident over a response time $\tau_r>5$~$\mu$s. Hence, the upper limit on the switch beam power threshold is 20~pW. Given how rapidly the response time increases as $P_s\rightarrow0$, the threshold is likely to be much lower. Additional data below $P_s=35$~pW would improve the estimates of these limits.

% subsubsection switching_photon_number (end)

\subsubsection{Transistor-like response} % (fold)
\label{ssub:transistor_like_response}

The response shown in Fig.~\ref{fig:datafull}, demonstrating the saturated and
linear response regimes, suggests that this device operates in a manner that
is analogous to an electronic transistor. Furthermore, the two response
regimes exhibited by the switch indicate that the output satisfies the
conditions for signal level restoration, as discussed in
Sec.~\ref{sec:criteria}.

For a device to exhibit signal level restoration, variations in the input
level cannot cause variations in the output level. In every device, however,
there is a narrow range of input levels, known as the intermediate region,
that lead to intermediate output levels. For input levels above or below the
intermediate range the output is \emph{saturated} as a logic high or low
respectively. In the case of the Rb-vapor device, this intermediate region is
between 900~pW and $<$35~pW. For input levels above 900 pW, the output is high
with a level set by the pump beam power.

Signal level restoration is a key property of the electronic transistor
enabling large networks of electronic logic elements. This demonstration of an
optical logic element that exhibits level restoration is a key step towards
practical optical switches. An all-optical transistor would have applications
in many data processing and communication networks in the future.

One notable limitation of the atomic-vapor switch system is the slow response time (on the order of 3~$\mu$s). A related approach, based on optical pattern formation in semiconductor systems, shows promise as a high-speed, high-bandwidth system. The details of nonlinear optical pattern formation in semiconductor systems, and a discussion of recent work, is presented in Section~\ref{sec:semiconductor_systems}.

% subsubsection transistor_like_response (end)

% subsection switch_response (end)

% section experiment (end)

\section{Numerical results} % (fold)
\label{sec:numerical_results}

In order to establish the necessary ingredients for a theoretical model to
describe the switching behavior observed experimentally, we develop a simple
extension to a previous model of a pattern-forming counterpropagating beam
system. Based on the model of Firth and Par\'e \cite{Firth_1988aa}, numerical
simulations performed by Chang \etal describe hexagonal pattern formation in a
counterpropagating beam system \cite{Chang_1992aa}. We extend this prior work
by simulating all-optical switching with transverse patterns. Specifically, we
simulate the effect of a weak switch beam on the orientation of the hexagonal
pattern generated by gaussian pump beams that counterpropagate through a
Kerr-type nonlinear medium. Simulations of the time response of this system
show behavior that is qualitatively similar to experimental observations.
In particular, the response time increases as the switch-beam power decreases.

\subsection{3D model} % (fold)
\label{sub:3d_model}

The model used in these simulations is described in Ref. \cite{Firth_1988aa}
and has been extended for our investigations to the case of two transverse
dimensions. We assume scalar fields, i.e., the model does not account
for the vector nature of the fields, and hence cannot describe polarization
instabilities, and we do not include absorption effects. Nonetheless, this
model is sufficient to describe pattern formation in counterpropagating-beam
nonlinear optical systems. The forward and backward fields counterpropagating
through a Kerr-like medium are described by the dimensionless
equations
\begin{align}
        \left(\frac{\partial}{\partial z}+\frac{\partial}{\partial t}\right)F&=\frac{i}{4\pi\mathcal{F}}\nabla_\perp^2F+i(|F|^2+2|B|^2)F,\label{eqn:model_eqns_F}\\
        \left(-\frac{\partial}{\partial z}+\frac{\partial}{\partial t}\right)B&=\frac{i}{4\pi\mathcal{F}}\nabla_\perp^2B+i(|B|^2+2|F|^2)B.\label{eqn:model_eqns_B}
\end{align}
Time is normalized by the transit time through the medium, $t_r=n_0 L/c$, the
longitudinal dimension $z$ is normalized by the medium length, and the
transverse dimensions $x,y$ are normalized by the beam waist $w_0$ where $\mathcal{F}$ is the Fresnel number, see Eq.~(\ref{eqn:fresnel_num}). $F$ ($B$)
is the forward (backward) field amplitude. The nonlinear coefficient $n_2$ is
scaled into the field amplitudes and is assumed to be positive as appropriate
for the experimental conditions described in
Sec.~\ref{sub:experimental_apparatus}. The medium length is also scaled into the field amplitudes such that $F^2=IL$ where $I$ is the pump-beam intensity. One consequence of this scaling is that the product $IL$ represents the nonlinear phase shift, in radians, experienced by an off-axis wave.

The transverse profile of the pump waves are assumed to be Gaussian such that
\begin{align}
  F(x,y,0,t) &= F_0 e^{-(x^2+y^2)}e^{i(K_x x + K_y y)} + \xi(x,y),\\
  B(x,y,L,t) &= B_0 e^{-(x^2+y^2)},
\end{align}
where $L$ is the medium length, $K_{x,y}$ sets the misalignment between
forward and backward wave-vectors, and $\xi$ is a delta-correlated Gaussian
random variable with $\langle\xi\rangle=0$. The random time-independent noise
source is included to simulate the effects of small spatial variations in the
input beam. Typically, the peak-to-peak noise amplitude is set to
$\Delta\xi=0.01$, equivalent to 1 percent of the pump field amplitude.

Equations~(\ref{eqn:model_eqns_F}) and (\ref{eqn:model_eqns_B}) are solved
numerically using a split-step beam propagation method where linear
diffraction is computed via fast Fourier transform \cite{Fleck_1976aa}. The
numerical grid of 256 by 256 transverse points and 20 longitudinal slices. The
numerical grid is slightly rectangular with the $x$ dimension 1\% larger than
the $y$ dimension. This prevents the square symmetry of the grid from biasing
the pattern formation process. Additionally, suitable choice of parameters and
appropriate spatial filtering are used in order to avoid spurious high-$K$
instabilities \cite{Firth_1990aa,Penman_1990aa}.

% subsection 3d_model (end)

\subsection{Controlled pattern rotation} % (fold)
\label{sub:controlled_pattern_rotation}

The primary result presented by Chang \etal is the formation of hexagonal
patterns in a three-dimensional model of gaussian beams counterpropagating
through a medium exhibiting Kerr nonlinearity. Their simulations are conducted
with $\mathcal{F}=63.7$ and $IL=0.565$, where the threshold for plane-wave
pattern formation predicted by Firth and Par\'e is $IL\simeq0.45$. Therefore,
Chang \etal simulate pattern formation for pump beams that are 25\% above the
minimum plane-wave threshold.

We have conducted simulations with a wide range of values of $\mathcal{F}$
between 64 and 4, where our experimental conditions correspond to
$\mathcal{F}=5.3$. Simulations in this range all exhibit hexagonal pattern
formation and reproduce the results of Chang et al. In order to
simulate the specific geometry of our experiment, the results reviewed here
are of simulations where $\mathcal{F}=5.3$ and $IL=0.565$ ($\sim25$\% above
threshold).

\begin{figure*}[htbp]
  
  \sidecaption
  \includegraphics*[scale=0.6]{Figure_7.eps}
  \caption{\label{fig:simframes} Numerical simulation of counterpropagating
gaussian beams shows ring and hexagon pattern formation in the far field. (a)
For this case, the pump beams are perfectly counter-propagating $K_x=K_y=0$.
The circle indicates the location of the switch beam in the final
three frames. (b) The location of the on- and off-state apertures are
indicated relative to the initial hexagon pattern that forms at $t=53$. The
on-state aperture (upper square) is located opposite the applied switch beam,
and the off-state aperture (lower square) transmits the spot immediately
counter-clockwise from the on-state aperture.}

\end{figure*}

Images of the far field pattern generated in a typical simulation are shown in
Fig.~\ref{fig:simframes}(a), where the time corresponding to each frame is
indicated in units of the transit time $t_r$. In the initial frame of
Fig.~\ref{fig:simframes}(a), the transmitted forward pump-beam is visible at
the center, and the weak off-axis perturbation is visible to the right. This
perturbation is used to quickly induce hexagonal pattern formation. Without
the initial perturbation, hexagons are spontaneously generated after 100-150
transit times. At $t=17$, the field that is conjugate to the perturbation, and
due to forward four-wave mixing, is visible to the left of the central pump.
The dark dot in the center of the first two frames is the result of numerical
filtering used to remove the DC artifact introduced by computing the far-field
via FFT.\footnote{This filtering is only performed on the images in order to
improve the contrast, and not during the simulation itself.} At $t=23$, a ring
pattern has formed that is replaced by hexagons at $t=53$. The seed beam is
turned off at $t=35$ and is not visible at $t=53$. It is interesting to note
that the ring pattern, predicted by generalizing the models of Yariv and
Pepper \cite{Yariv_1977aa} or Firth and Par\'e \cite{Firth_1988aa} to
cylindrically symmetrical transverse dimensions is a transient solution that
appears early ($t=23$) in the development of the off-axis patterns. The ring
is not a stable solution for the system in the presence of symmetry breaking,
due in this case to the initial seed beam, and the ring breaks up into six
spots after a short time. The second row of frames shown in
Fig.~\ref{fig:simframes}(a) are collected after the application of an off-axis
switch-beam, which turns on at $t=85$, and are discussed in the next
section.

% subsection controlled_pattern_rotation (end)

\subsection{Switch response} % (fold)
\label{sub:num_switch_response}

In our simulations, much like in the experiments of Dawes \etal
\cite{Dawes_2008aa}, we observe that injecting a weak switch beam into the
nonlinear medium after hexagons have formed causes the hexagon pattern to
rotate such that a bright spot is aligned to the direction of the switch beam.
This rotation is illustrated in the lower four frames of
Fig.~\ref{fig:simframes}(a). The switch beam is applied at $t=85$, and becomes
visible between the two right-side spots at $t=101$. For the frames shown, the
switch beam power is $P_s=10^{-4}P_p$, where $P_p$ is the power of each of the
counterpropagating pump beams. At $t=150$, the counterclockwise rotation of
the pattern can be observed and continues until the end of the simulation at
$t=300$ where the pattern has rotated such that the locations that were
previously bright are now dark.

As in the experiments, the patterns generated in this simulation can be
spatially filtered in order to define two or more output channels.
Figure~\ref{fig:simframes}(b) indicates the location of the apertures used to
filter the numerical results. Square apertures are used for numerical
efficiency, but the results are not expected to differ if they are replaced
with round apertures. The power transmitted by these apertures is calculated
by summing the simulated intensity values within each aperture. For four
simulation runs, each with different switch-beam power, the power transmitted
through the on- and off-state aperture as a function of time $t$ is shown in
Fig.~\ref{fig:simresponse}(a) and (b), respectively.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figure_8.eps}
  \end{center}
  \caption[The power transmitted by simulated apertures.]{The power
  transmitted by apertures in the numerical model exhibits switching behavior
  that is similar to the experimental system. The response of the on- and
  off-state aperture transmission is shown for four levels of switch-beam
  power. The switch beam is turned on at $t=85$, indicated by the arrow in
  (a). As the switch-beam power decreases, the simulation exhibits slower
  response, i.e., slower pattern rotation. The switch-beam power (in
  units of pump-beam power) corresponding to these four traces are $10^{-4}$
  (solid black), $2.5\times10^{-5}$ (large dash blue), $4\times10^{-6}$ (small
  dash red), and $1\times10^{-6}$ (dash-dot green). The horizontal dotted line
  indicates the threshold used to calculate response times for the simulated
  switch.}
  \label{fig:simresponse}
\end{figure}

After the initial transients in the pattern formation, the power in the off-
and on-spots stabilize within 50 transit times. At $t=85$, the switch-beam
is applied and the pattern begins to rotate, transferring power from the
off-state aperture to the on-state aperture. For $P_s=10^{-4}P_p$, complete
rotation occurs within 200 transit times. For lower switch beam power, the
pattern rotates more slowly as the remaining traces show in
Fig.~\ref{fig:simresponse}. To compare the change in response time observed
in the simulation to that observed experimentally, we measure the response
time of the simulated switch as the time between the application of the switch
beam ($t=85$) and the threshold crossing for the on-spot. The threshold,
also shown in Fig.~\ref{fig:simresponse}, is chosen to roughly correspond to
the threshold level used in the experiments.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figure_9.eps}
  \end{center}
  \caption[Simulation of the switch response time.]{Simulation of the switch exhibits an increase in response time for decreasing power that is qualitatively similar to experimental observations. To facilitate comparison to Fig.~\ref{fig:responsetime}(a), the horizontal axis has high switch-beam power to the left and low switch-beam power to the right.}
  \label{fig:simresponsetime}
\end{figure}

The response time of the simulated switch data shown in
Fig.~\ref{fig:simresponsetime} ranges from 40 transit times to 210 transit
times, as shown in Fig.~\ref{fig:simresponsetime}. For comparison, the
transit time of the 5-cm-long vapor cell used in our experiment is 160~ps, so
the simulated response times would correspond to experimental values of 6.4~ns
and 33.6~ns respectively. Experiments observe response times between 2 and
4~$\mu$s in Rb vapor, so it is clear that this numerical model does not
agree quantitatively with these observations. However, the simulated
response time does exhibit a sharp increase in the limit of low switch-beam
power, which is qualitatively similar to experimental observations. This
increase in response time for weak inputs may be an indication that the switch
undergoes critical slowing down \cite{Strogatz_2001aa}, which would not be
surprising since the orientation of the pattern exhibits multi-stability
between the preferred orientations.

Another notable feature of these numerical results is that, despite the
limitations of the model, the amount of switch-beam power, relative to the
total pump power, required to rotate the pattern is of the same order of
magnitude as what has been observed experimentally. For reference,
Table~\ref{tab:exp_sim} shows the correspondence between the normalized
switch-beam power $P_s/P_p$ used in the simulations presented above and the
experimental values, based on total pump-beam power of $P_p=560~\mu$W from the
experiments in Ref.~\cite{Dawes_2008aa}. As an example, the third curve in
Fig.~\ref{fig:simresponse} corresponds numerically to $P_s=4\times10^{-6}
P_p$. In the work of Dawes et al., this switch-beam to pump-beam power ratio
would imply a switch-beam power of 1~nW and their switch typically operates
between 1~nW and 50~pW. Therefore, the sensitivity demonstrated in
experimental work is largely described by this model.

\begin{table}
  \begin{center}
  \begin{tabular}{c|c}
    $P_s/P_p$&$P_s$[nW]\\
    \hline\vspace{0.1cm}
    $1\times10^{-4}$&26\\
    $2.5\times10^{-5}$&6.5\\
    $4\times10^{-6}$&1\\
    $1\times10^{-6}$&0.26\\
  \end{tabular}
  \end{center}
  \caption[Correspondence between $P_s/P_p$ and $P_s$ in nW]{The
  correspondence between $P_s/P_p$ and $P_s$ in nW based on 560~$\mu$W of
  total pump power.}
  \label{tab:exp_sim}
\end{table}

There are certainly features of the experiment that these simulations do not
capture. In the first case, absorption is neglected in assuming a Kerr-type
nonlinearity. One consequence of this is that misalignment of the pump beams
in the simulation does not serve to reduce the number of spots generated. This
is in contrast to the experiment, where misaslignment of the pump beams
results in a pair of spots rather than a hexagon. This is likely due to the
fact that, without simulating absorption, there is no loss experienced by the
less-favored hexagonal components and, even for large pump-beam misalignment,
the pattern remains a hexagon. Simulations that include misalignment of the
forward pump beam exhibit hexagonal pattern formation in addition to
fluctuations in the pattern orientation and a near-field pattern flow
\cite{Honda_1995aa}. Just as for well-aligned pump beams, the switch beam also
causes pattern rotation when the forward pump beam is misaligned, and the
switch response time diverges near zero switch-beam power in the misaligned
case as well.

Symmetry breaking may be responsible for pinning the orientation of the
pattern. Including pump-beam misalignment in these simulations does appear to
have this effect and is the focus of ongoing work. Refinement of the model to
include absorption and saturation may also improve the agreement between
experiment and simulation. Furthermore, because we have assumed a medium
with an instantaneous nonlinear response, the only timescale in the Kerr model
is the transit time. This leads to significantly faster switch response in the
model compared to experimental observations. To quantitatively model the
response time requires a more refined model of the nonlinear interaction that
includes optical pumping effects and the associated time scales.

Although these initial simulations of pattern-based all-optical switch exhibits a fast response time, the experimental implementation in atomic vapor is relatively slow, and hence is a low-bandwidth system. Recent work has explored the possibility of extending these initial results by developing semiconductor systems that exhibit nonlinear optical pattern-formation. The final section describes nonlinear optics in a semiconductor system, and reviews recent results that demonstrate that such systems are potential candidates for high-bandwidth all-optical devices.

% subsection switch_response (end)

% section numerical_results (end)

\section{Semiconductor systems} % (fold)
\label{sec:semiconductor_systems}



The promising experimental results on pattern switching in atomic vapor
(gaseous) systems have raised the question whether similar effects
can be expected in solid state systems, in particular
semiconductors. One obvious advantage of semiconductors over atomic
systems would be the fact that they can be more easily integrated in
optoelectronic communications networks. Currently, many
semiconductor devices are based on III-V compounds, such as GaAs,
but beyond that there is a large variety of other systems, from
II-VI compounds to group-III-nitride materials to zinc oxide
materials to silicon  structures. Quite generally, semiconductor
systems offer great flexibility in terms of epitaxial system growth
(including active layers and mirrors), and, of course, they are
mechanically robust.

While those application aspects suggest that semiconductors can be useful
alternatives to the gaseous systems described in the previous sections, one
needs to realize that the physics underlying the optical nonlinearities and
the resulting optical instabilities are very different in semiconductors
compared to gases. As we will show below, not all of these differences favor
the semiconductor system. From a general point of view, we note that most
semiconductor-based optoelectronic devices operate at frequencies close to the
fundamental bandgap energy, i.e., they operate either close to an exciton (the
exciton being a bound electron-hole pair) resonance or even in the
band-to-band continuum. In order to achieve optical instabilities, the optical
nonlinearity needs to be sufficiently large. Generally, this can be achieved
by tuning the pump beams close to an optical resonance and by using
sufficiently high intensities. We have seen in the previous sections that, in
the atomic case, quasi resonant excitation (within the Doppler-broadened
atomic spectral line) and high pump intensities (leading to significant
bleaching of the line) created the optical instabilities. It is then natural
to ask whether it is possible to create similar instabilities in a
semiconductor. Excitation near the lowest exciton resonance and with
sufficiently high intensity can create similar instabilities. However, under
high excitation, an atomic resonance behaves very differently from an exciton
resonance. In atoms, strong pumping can bleach the resonance and also lead to
hole-burning and ac Stark shifts. In these cases, the nonlinearity can often
be modeled, albeit approximately, by a single parameter $n_2$. This is in
sharp contrast to excitons, where optical excitation creates a complex
many-particle system that fundamentally changes the physics of the transition.
For example, in the lowest-order nonlinear optical regime (the
$\chi^{(3)}$-regime), optical excitation creates, among other things,
two-exciton Coulomb correlations. Depending on the vectorial polarization
state of the optical beams, these correlations may include bound two-exciton
states (biexcitons). The biexciton resonance and the two-exciton continuum
correlations lead to strong excitation-induced dephasing (EID), which is
usually much larger than the corresponding contribution from PSF (compare Fig.
\ref{Tpp.fig}b discussed below). In the language appropriate for
semiconductors, the latter is associated with phase-space filling (PSF)
\cite{kwong-etal.05}. The fact that in a semiconductor the optical pump beam
creates carriers that lead to increased dephasing rates (EID) makes it
generally more difficult to achieve optical instabilities. It therefore makes
it necessary to study the origins of optical nonlinearities and optical
instabilities in semiconductors in detail in order to provide a path toward
pattern formation and all-optical switching. It is necessary to find parameter
values and configurations in which the instability threshold intensity is kept
small. For example, EID has a strong detuning dependence in the vicinity of
the exciton resonance (where by detuning we mean the difference between the
center frequency of the optical pump field, $\hbar \omega_p$ and the exciton
resonance $\varepsilon_x$, $\Delta \varepsilon = \hbar \omega_p -
\varepsilon_x$). The problem of EID can be controlled (or to some degree
engineered) with the help of quantum confinement and cavity enhancement, as
detailed below. Also, the optical nonlinearities depend critically on the
intensity and vectorial polarization of the pump beam. Depending on the
precise light beam and material parameters (including the system's geometry
that may or may not include cavity mirrors), the nonlinearity can be dominated
by PSF effects, instantaneous Hartree-Fock (HF) Coulomb effects, and
time-retarded two-exciton correlations, to name the ones that will be
discussed in more detail later.




\subsection{Excitonic optical nonlinearities}

Absorption spectra of non-excited direct-gap semiconductors exhibit
discrete excitonic resonances, typically a few meV below the
fundamental bandgap. Nonlinear processes involving excitons have
been the subject of intensive research for several decades (for
recent text book treatments and reviews see, for example
\cite{haug-jauho.96,chemla-shah.01,schaefer-wegener.02,haug-koch.04,meier-etal.06}).
In semiconductor amplifiers and lasers we do not have discrete
excitonic resonances, but continuous spectral gain and absorption
regions. In the following, we will focus solely on
excitonic resonances, since they have been studied in great detail
and their optical nonlinearities are by now well understood.
Furthermore, because of their dominant role in optoelectronic device
concepts, we will discuss semiconductor quantum-well systems, i.e.,
quasi-two dimensional systems. In thin GaAs quantum wells, the
lowest optical transitions are dominated by heavy-hole excitons. In
this case, the optical dipole selection rules are particularly
simple. Both the heavy-hole valence and the conduction band are
two-fold degenerate, and right circularly polarized light (denoted
by ``+'') couples one valence band with one conduction band, whereas
left-circularly polarized light (denoted by ``-'') couples the other
two bands.


In order to discuss excitonic optical instabilities, it is
advantageous to analyze the equation of motion of the excitonic
interband polarization $p(t)$ separately from the Maxwell
propagation equation. Similar to the instabilities discussed in Sec.
\ref{sub:3d_model}, one can formulate the nonlinear equation of
motion for $p(t)$ in a way that generalize the concept of
phase-conjugate oscillation (PCO) \cite{Yariv_1977aa}. Before
describing the optical instabilities, we first discuss briefly the
physical origin of the excitonic nonlinearities. If a pump beam with
frequency close to the lowest exciton resonance creates many
excitons in the system, the constituent electrons and holes interact
via the Coulomb interaction, and in addition the Pauli principle
yields PSF effects. We assume here that the intensity of the pump
beam is not too high, so as to avoid exciton ionization and the
formation of an electron-hole plasma. The effects of the Coulomb
interaction are usually divided into (static) HF interactions and
correlation effects. In the lowest-order (in the light field
amplitude) nonlinear regime, and if the optical excitation contains
both circular polarizations ``+'' and ``-'', the two-exciton
correlations contain bound two-exciton states (biexcitons) as well
as two-exciton scattering continua. If the optical excitation
contains only one circular polarization (either {``+''} and
{``-''}), the correlations contain only two-exciton continua. All
two-exciton correlations can contribute to EID, as will be shown
below. The equation of motion for the coherent excitonic interband
polarization is
\cite{Axt1994a,Oestreich1995,kwong-etal.01prb,Takayama2002}
\begin{align}\label{p-motion}
i\hbar\dot{p}^\pm&=(\varepsilon_x-i\gamma)p^\pm\nonumber\\&-\big[\phi^{\ast}_{1s}(0)-2A^{\text{PSF}}|p^\pm|^2\big]d_{cv}E^\pm+V^{\text{HF}}|p^{\pm}|^2p^\pm 
\nonumber \\&+
2p^{\pm\ast}\int_{-\infty}^{\infty}{\mathrm{d}\,t'\mathcal{G}^{\pm\pm}(t-t')p^\pm(t')
p^\pm(t')} \nonumber \\ &+
p^{\mp\ast}\int_{-\infty}^{\infty}{\mathrm{d}\,t'\mathcal{G}^{\pm\mp}(t-t')p^\mp(t')
p^\pm(t')}\,.
\end{align}
Here, $\varepsilon_x$ is the 1s-hh exciton energy, $\gamma$ a
phenomenological excitonic dephasing constant, $d_{cv}$ the
interband dipole matrix element and $E$ the light field amplitude at
the position of the QW. Both $p$ and $E$ depend on time and on the
coordinate vector ${\bf r}=(x,y)$ in the plane of the QW.
$A^{\text{PSF}}=4a_0^x\sqrt{2\pi}/7$ accounts for excitonic PSF,
where the bulk exciton Bohr radius is denoted by $a_0^x\approx170\,\AA$. The
two-dimensional 1s exciton wavefunction $\phi_{1s}(\mathbf{r})$ is
evaluated at $\mathbf{r}=0$.
$V^{\text{HF}}=2\pi(1-315\pi^2/4096)/a_0^{x2}E_b^x$ (with the
exciton binding energy $E_b^x\approx13\,\mathrm{meV}$) is the HF
Coulomb matrix element. Unless otherwise noted, the time argument is
$t$.  The correlation kernels $\mathcal{G}$ are given by
$\mathcal{G}^{++}=\mathcal{G}^{--}=\tilde{G}^+$ and
$\mathcal{G}^{+-}=\mathcal{G}^{-+}=\tilde{G}^++\tilde{G}^-$, with
$\tilde{G}^{\pm}$ as defined in Eq.~(22) of
Ref.~\cite{Takayama2002}, including a two-exciton dephasing rate
$2\gamma$. In Eq. (\ref{p-motion}), we have neglected excitonic
correlation of order 3 and higher, since such correlations are
usually weak and very difficult to detect \cite{Axt2004}.

In order to illustrate the physical contents of the various nonlinear terms in
Eq. (\ref{p-motion}), we Fourier transform the correlation functions from the
time domain to the frequency ($\Omega$) domain and show their frequency
dependence in Figs. \ref{Tpp.fig} and \ref{Tpm.fig}. In Fig. \ref{Tpp.fig}, we
also include the effect of PSF. The corresponding $G^{\text{PSF}}$ follows
from the PSF term in Eq. (\ref{p-motion}) if $E$ is expressed in terms of the
first-order $p$ \cite{kwong-etal.01prb}. In pump-probe configurations
(including the instability analysis discussed in the next subsections), the
real parts of the $G$'s are proportional to the pump-induced shift of the
exciton resonance, while their imaginary part is a measure of EID. Consistent
with Ref. \cite{kwong-etal.01prb}, we call $T^{++} = V^{HF}+2\mathcal{G}^{++}$
and $T^{+-} = 2\mathcal{G}^{+-}$ the T-matrix in the ``++'' and ``+-''
channel, respectively. We see from Fig. \ref{Tpp.fig}a that, in a system in
which there is only one circular polarization, HF yields a blue shift that
overcompensates the correlation-induced red shift. PSF yields a shift that is
positive (negative) below (above) the two-exciton continuum edge ($\hbar
\Omega = 2 \varepsilon(0)$). Adding the PSF to the HF contribution yields a
negative slope on the HF blue shift with zero contribution at the two-exciton
continuum edge. Figure~\ref{Tpp.fig}b shows that, in the two-exciton continuum
($\hbar \Omega > 2 \varepsilon(0)$), the two-exciton correlations yield strong
EID, whereas the correlation-induced EID becomes negligible below the
two-exciton continuum. Furthermore, PSF yields a small contribution to EID
independent of frequency.



\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=0.45,trim=0 00 0 00]{Figure_10.eps}
  \end{center}
  \caption{Exciton-exciton T-matrix in the co-circular polarization channel with $\gamma=0.75$~meV. Here, $\varepsilon(0)$ denotes the 1s-hh exciton energy.
  From Ref.~\cite{kwong-etal.01prb}.}
  \label{Tpp.fig}
\end{figure}



\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=0.45,trim=0 00 0 00]{Figure_11.eps}
  \end{center}
  \caption{Same as Fig. \protect\ref{Tpp.fig}, but for the counter-circular polarization channel. From Ref.~\cite{kwong-etal.01prb}.}
  \label{Tpm.fig}
\end{figure}



Figure~\ref{Tpm.fig}a shows that, in the counter circular polarization
channel, we have the biexciton resonance below the two-exciton continuum.
Here, the shift has a cross-over from red to blue. The biexciton resonance
also yields strong EID, as can be seen in Fig. \ref{Tpm.fig}b. In these
figures, a relatively large dephasing has been used. For smaller values of
$\gamma$, the spectral region where biexcitonic EID is large becomes narrower
\cite{Takayama2002}.

Knowledge of the frequency dependent excitonic PSF, HF and
correlation effects is crucial for the search of optical
instabilities and pattern formation. Using, as general guidelines,
the criteria that a shift towards the pump frequency (which reduces
the effective detuning and thus enhances the action of the pump
beam) and small EID is beneficial for instabilities, we conclude
from Figs. \ref{Tpp.fig} and \ref{Tpm.fig} that, in the co-circular
channel, pumping above the two-exciton continuum yields the desired
shift but unfortunately strong EID, whereas pumping below the
two-exciton continuum  avoids EID but unfortunately yields a HF
shift of the exciton resonance away from the light frequency. We
will see in Sec. \ref{microcavities.sec} that use of a microcavity
can yield a way out of this dilemma. This is because the EID
increases with increasing effective mass (here the mass of the
exciton). As we will discuss in more detail below, a small polariton
mass in a microcavity can substantially reduce EID.

In the counter-circularly polarized channel, Fig. \ref{Tpm.fig}a
does not give clear guidelines for possible instabilities, but one
might assume that the region around the biexciton resonance may
yield the desired sign of the light-induced exciton shift and small
EID if the biexcitonic dephasing is sufficiently small. We will
explore this possibility in the next section.



\subsection{Instabilities in single quantum wells}

In order to study instabilities in single quantum wells, we assume the geometry depicted in Fig. \ref{fig:sketch-qw}, with a pump beam in normal incidence and a probe beam at a small angle relative to normal incidence. The exciton polarization is restricted to the quasi-two dimensional plane of the QW.
The polarization component corresponding to the
background-free four-wave mixing direction, denoted in the following by the subscript $f$, travels in the direction
${\bf k}_f= 2{\bf k}_p  -{\bf k}_s =-{\bf k}_s $, since ${\bf k}_p= 0$ (where all wavevectors are two-dimensional vectors).
Hence, the two-dimensional spatial Fourier decomposition of the excitonic polarization, with Fourier components up to first order in the grating wavevector, yields
\begin{align} \label{FTdecomp.equ}
p^{\pm}(t, {\bf r}) & = & p_p^{\pm}(t) + p_s^{\pm}(t) e^{ i {\bf
k}_s {\bf r}} + p_f^{\pm}(t) e^{ - i {\bf k}_s {\bf r}}\,.
\end{align}
In these systems, counterpropagating beams are not needed for
backward four-wave mixing and instabilities of the PCO type. One
cannot distinguish
 between forward four-wave mixing and backward four-wave mixing.
\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=0.6]{Figure_12.eps}
  \end{center}
  \caption{Geometry of the four-wave mixing process in a semiconductor quantum well (QW). The in-plane wave vectors of the interband polarization are indicated, with the pump wavevector being zero.}
  \label{fig:sketch-qw}
\end{figure}

Because biexcitonic effects can be assumed to be critical for optical
instabilities \cite{Peyghambarian1983,Inoue1986,Koch1981,Nguyen1994}
in single quantum wells, it is advantageous to re-write
Eq.~(\ref{p-motion}) in a way that allows for a linear stability
analysis with full inclusion of the temporal retardation effects
related to biexciton formation \cite{Sieh1999}. This can be achieved
by separating the continuum part and the bound biexciton part in the
correlation function $\mathcal{G}^{\pm\mp} =
\mathcal{G}_{\text{cont}}^{\pm\mp}+ \mathcal{G}_{xx}^{\pm\mp}$ and
eliminating $\mathcal{G}_{xx}^{\pm\mp}$ in favor of an equation of
motion for the biexciton amplitude $b(t)$, yielding new equations of motion
\begin{align}\label{p-motion-2}
i\hbar\dot{p}^\pm&=(\varepsilon_x-i\gamma)p^\pm \nonumber \\
&-\big[\phi^{\ast}_{1s}(0)-2A^{\text{PSF}}|p^\pm|^2\big]d_{cv}E^\pm +V^{\text{HF}}|p^{\pm}|^2p^\pm \nonumber \\ 
&+2p^{\pm\ast}\int_{-\infty}^{\infty}{\mathrm{d}\,t'\mathcal{G}^{\pm\pm}(t-t')p^\pm(t')p^\pm(t')} \nonumber \\
&+p^{\mp\ast}\int_{-\infty}^{\infty}{\mathrm{d}\,t'\mathcal{G}_{\text{cont}}^{\pm\mp}(t-t')p^\mp(t')p^\pm(t')} \nonumber \\
&+ C_{xx}^{\ast} p^{\pm\ast} b(t)\,,
\end{align}
with
\begin{align}\label{b-motion}
i\hbar&\dot{b} = (\varepsilon_{xx} - 2 i \gamma) b + \frac{1}{2}
C_{xx} p^+ p^-\,,
\end{align}
where $C_{xx}$ is a function of the exciton-exciton interaction and
the biexciton wavefunction, and is taken to be $0.54E_b^x a_0^x$ in
the following. The propagation of the optical field $E^\pm$ across
the QW is described with a transfer-matrix method that accounts for
radiative corrections and that assumes the QW to be infinitely thin
(see, for example, Eq. (A5) of Ref. \cite{kwong-etal.01prb}). It is
important to note that, in a single quantum well, the excitonic
polarizations are sources for light fields via radiative decay, but
there is no feedback of the radiative decay on the incoming light
fields. Hence, the dynamics of the system can be described solely
with the equation for the excitonic polarizations. The situation
will be different in the microcavity (Sec. \ref{microcavities.sec}),
where the strong interaction between the excitonic polarization and
the cavity mode require a full description of the system dynamics
including the equations for $p$ and $E$.




Using the Fourier decomposition (Eq. (\ref{FTdecomp.equ})) of Eqs.
(\ref{p-motion-2}) and (\ref{b-motion}), it is straighforward to
derive the linear set of equations of motion for $p_s^\pm$ and
$p_f^\pm$ (fully given in Ref. \cite{Schumacher2006}) and the
nonlinear equation for the pump-induced interband polarization
$p_p^\pm$, which is independent of $p_{s,f}^\pm$. The equations for
$p_{s,f}^\pm$ contain self-wave mixing (SWM) terms,
 $i\dot{p}_{s,f}\sim
p_{s,f}p_p^\ast E_p $
and
$i\dot{p}_{s,f}\sim p_{s,f}p_p^\ast p_p $,
and cross-wave mixing (XWM) terms,
$i\dot{p}_{s,f}\sim p_{f,s}^\ast p_pE_p$
and
$i\dot{p}_{s,f}\sim p_{f,s}^\ast p_p p_p $. The XWM terms are necessary for instabilities and possible pattern formation.






To analyze the possibility of optical instabilities, we perform a linear stability analysis (LSA). The LSA is done
without an incoming probe field and for a monochromatic cw pump
field $E^\pm_{p}(t)=\tilde{E}^\pm_{p}\mathrm{e}^{-i\omega_pt}$ and
pump polarization
$p^\pm_{p}(t)=\tilde{p}^\pm_{p}\mathrm{e}^{-i\omega_pt}$, with
$\dot{\tilde{p}}^\pm_p=\dot{\tilde{E}}^\pm_p=0$ ($\omega_p$ is the
pump frequency). We evaluate the memory integrals using the Markov approximation [$p_{s,f}(t')\approx
p_{s,f}(t)\mathrm{e}^{i\omega_p(t-t')}$] for the two-exciton
continuum in the correlation kernels $\mathcal{G}^{\pm\pm}$,
$\mathcal{G}^{\pm\mp}$. The term driving the bound biexciton
amplitudes $b_{s,f}(t)$ is proportional to $p_p^\mp
p_{s,f}^\pm+p_p^\pm p_{s,f}^\mp$. With the ansatz
$p_{s,f}(t)=\tilde{p}_{s,f}(t)\mathrm{e}^{-i\omega_pt}$ and
$b_{s,f}(t)=\tilde{b}_{s,f}(t)\mathrm{e}^{-i2\omega_pt}$, the probe
and FWM dynamics take the form
\begin{align}\label{LSA.equ}
\frac{\mathrm{d}}{\mathrm{d}t}{\tilde{\mathbf{p}}}(t)=M\,{\tilde{\mathbf{p}}}(t),
\end{align}
with ${\tilde{\mathbf{p}}}(t)=\big[{\tilde{p}}^+_s(t),
{\tilde{p}}_f^{+\ast}(t), {\tilde{p}}^-_s(t),
{\tilde{p}}_f^{-\ast}(t), \tilde{b}_s(t),
\tilde{b}_f^{\ast}(t)\big]^T$. The system is unstable if at least one of the eigenvalues $\lambda_{i}$ of $M$ fulfills $\mathrm{Re}\{\lambda_{i}\}>0$.

Since atomic nonlinearities are clearly very successful in terms of allowing
instabilities, it is instructive to first evaluate Eq. (\ref{LSA.equ}) for the
semiconductor quantum well keeping only the atomic nonlinearities (PSF). For
this case, an analytical evaluation using first-order (in the pump field
amplitude) pump polarizations has been discussed in
Ref.~\cite{Schumacher2008a}. It was found that instability driven only by PSF
cannot be expected. However, a generalization of the model, in which spatial
dispersion of the exciton frequency is taken into account (i.e.,
$\varepsilon_x$ entering the pump equation is different from that entering the
signal and FWM equations) allows, at least in principle, for instabilities.
This is shown in Fig.~\ref{psf-with-dispersion.fig}. We show the case of
no-dispersion as well as two different in-plane wavevectors of the signal.
Here, $k_{\text{max}}$ is the maximum in-plane vector for the given frequency.
Clearly, we find a region of positive eigenvalues (i.e., instabilities) for
the case of $k_s=0.75k_{\text{max}}$, which indicates that spatial dispersion
is indeed beneficial of the PSF-driven instability.



\begin{figure}[t]%
\includegraphics*[width=\linewidth]{Figure_13.eps}
\caption{%
\label{psf-with-dispersion.fig} Linear stability analysis for a
linearly polarized pump for steady-state total coherent exciton
density $n_x^{\text{total}}=1.7\times10^{11}\mathrm{cm^{-2}}$. Shown
are the real parts  of the eigenvalues $\lambda_{i}$ of the matrix
$M$ vs. pump detuning. Only phase space filling (no exciton interaction) is included here. The dotted line separates the
stable (${\rm Re}\{\lambda\}<0$) from the unstable (${\rm
Re}\{\lambda\}>0$) regime. From Ref.
\protect\cite{Schumacher2008a}.}
\end{figure}


We stress that Fig. \ref{psf-with-dispersion.fig} is only a case study, meant
to illustrate how PSF could in principle yield near-resonance instabilities in
semiconductor quantum wells. However, PSF can only yield instabilities at
positive detuning, at which correlation processes yield large EID. In the
calculation leading to the result in Fig. \ref{psf-with-dispersion.fig}, we
had to use a relatively high density, $1.7\times10^{11}\mathrm{cm^{-2}}$; a
density where our exciton-only model may no longer be valid. Furthermore, even
if excitons were still dominating in that density, EID should be expected to
be very significant. It is therefore necessary to study the complete system,
including PSF, HF, continuum correlations and biexcitonic correlations.
Solving for this case Eq. (\ref{LSA.equ}), we find the instabilities depend
sensitively on the vectorial polarization of the pump beam. While, in this
case, for circularly polarized pump beams, we find no instabilities, linearly
polarized (say ``X'' polarized) pump beams do yield instabilities, as shown in
Fig.~\ref{eigenvalues.fig}. We find three different unstable regions
($\mathrm{Re}\{\lambda_i\}>0$) caused by the biexcitonic ($+-$) XWM terms. The
labels XX and XY denote the vectorial polarizations of the pump (always X) and
the unstable modes or probe fluctuations (either X or Y). We see that a pump
beam tuned into the two-photon resonance with the biexciton allows for a
polarization instability (the unstable probe is Y polarized), whereas in the
spectral region just below and above the biexciton resonance we have
polarization-preserving instabilities. The XY instability and the XX
instability below the biexciton are single-color instabilities (the imaginary
parts of the eigenvalues are degenerate in the instability region), whereas
the XX instability above the biexciton is a two color instability. In the
latter, the imaginary parts of the ``unstable'' eigenvalues are split, which
means that we have two modes with the same growth rates but different
frequencies. We note that at the density used in Fig.~\ref{eigenvalues.fig}, $1.6\times10^{-10}\mathrm{cm^{-2}}$, density-induced ionization of excitons can be safely assumed to be irrelevant.

While the single quantum well instabilities discussed so far could
possibly be used for pattern formation, it needs to be noted that
these instabilities are found to be relatively fragile. They can be
expected only if the dephasing rate is very small, such as that
reported in \cite{langbein-etal.01}. In the next subsection, we turn
our focus on a less fragile instability in a different semiconductor
system.





\begin{figure}[h]%
\includegraphics*[width=\linewidth]{Figure_14.eps}
\caption{\label{eigenvalues.fig} Linear stability analysis for a
linearly polarized pump for steady-state total coherent exciton
density $n_x^{\text{total}}=1.6\times10^{10}\mathrm{cm^{-2}}$. Shown
are the real parts (a) and imaginary parts (b) of the eigenvalues
$\lambda_{i}$ of the matrix $M$ for negative pump detuning.
Eigenvalues are represented by solid lines for the co-linear (XX)
and by dashed-dotted lines for the cross-linear (XY) polarization
configuration. From Ref. \protect\cite{Schumacher2006}.}
\end{figure}





\subsection{Pattern switching in semiconductor microcavities}
\label{microcavities.sec}



\begin{figure}[h]%
\includegraphics*[scale=0.4,trim=0 350 0 0]{Figure_15.eps}
\caption{\label{sketch-cavity.fig} Sketch of a planar semiconductor quantum well (QW) microcavity, with distributed Bragg reflector (DBR) mirrors and normal-incidence pump beam.}
\end{figure}

In the following, we consider planar semiconductor microcavities,
which consist of a semiconductor quantum well and two mirrors (Fig.
\ref{sketch-cavity.fig}). In high quality cavities, the
exciton-photon coupling is strong enough so that the eigenmodes of
the system become cavity polaritons \cite{Khitrova1999}. The
in-plane dispersion of these polaritons is shown in Fig.
\ref{dispersion} along with the dispersions of the uncoupled
excitons and photons. The parabolic dispersion of the uncoupled
excitons is not visible in this figures because of the small
wavevector region shown. It is apparent from the figure that the
effective mass of the lower polariton branch (LPB) is much smaller
than that of the uncoupled exciton. As mentioned above, small mass
is related to small EID \cite{Kwong2001b,Savasta2003}, and the
smallness of EID at the LPB has been a major factor for their
usefulness in providing optical instabilities.



In the past decade, the parametric amplification of polaritons  (a
process usually related to optical instability) has been the subject
of intense experimental and theoretical research; see, e.g.,
Refs.~\cite{Savvidis2000,Huang2000,Ciuti2000,Stevenson2000,Saba2001,Whittaker2001}
or the reviews given in
Refs.~\cite{Ciuti2003,Baumberg2005,Keeling2007}. In a typical
pump-probe setup in a co-circular polarization configuration (with
the pump coming in at an angle $\vartheta$, unlike the case shown in
Fig. \ref{sketch-cavity.fig}), the amplification of a weak probe
pulse at normal incidence has mainly been attributed to four-wave
mixing (FWM) processes mediated by the repulsive Coulomb interaction
of the exciton constituent of the polaritons excited on the lower
polariton branch (LPB)
\cite{Ciuti2000,Stevenson2000,Whittaker2001,Savasta2003}. For a
specific pump in-plane momentum (defining the so-called ``magic
angle''), energy and momentum conservation is best fulfilled for the
FWM processes and thus a pronounced angular dependence of this
amplification is observed \cite{Savvidis2000,Ciuti2000}. Because, in
the strong coupling regime, the LPB is spectrally well below the
two-exciton scattering continuum, the influence of excitonic
correlations in the scattering processes of polaritons on the LPB is
strongly suppressed (compared to the situation in a single quantum
well  without the strong coupling to a confined photon cavity mode,
as discussed in the previous subsection). However, even for
co-circular pump-probe excitation, these correlations must be
considered for a complete understanding of the experimental results
\cite{Saba2001,Savasta2003,Savasta2003b}.





\begin{figure}[t]
{\begin{center}
\includegraphics[scale=0.95]{Figure_16.eps}
\end{center}}
\caption{\label{dispersion}(a) Sketch of the linear cavity polariton
dispersion. The bare cavity and exciton dispersions are shown,
together with the lower (LPB) and upper (UPB) polariton branches of
the coupled cavity-mode exciton system. The fundamental pairwise
off-axis scattering of pump polaritons is also indicated. (b) Sketch
of the hexagonal switching geometry in the transverse plane. The
elastic circle is defined by the pump frequency and the dispersion
of the LPB. The basic switching action triggered by the probe is
indicated. The radial bars indicate the variation in the magnitude
of off-axis momenta $\mathbf{k}$ as included in the nonlinear
polariton dynamics. From Ref. \protect\cite{Schumacher2007e}.}
\end{figure}


\begin{figure}[t]
\begin{center}
\includegraphics[scale=1.0]{Figure_17.eps}
\caption{\label{Figswitch}(a)-(c) Switching in the
\textcolor{black}{output signals in a reflection geometry (the
signals with out-of-plane momentum opposing the incident pump's are
plotted)}. The intensities per direction are normalized to the
incoming control intensity. The switching signal in (b) is about
$15$ times stronger than the incoming control in (d) that is
triggering this signal (note the different scales on the vertical
axes in panels (a)-(c) and (d)). In panel (b), direction 2 is shown
as the solid line and direction 5 as the dashed line. Similar
switching is observed in \textcolor{black}{a transmission geometry}
(not shown). From Ref. \protect\cite{Schumacher2007e}.}
\end{center}
\end{figure}


In the following, we review the recent work
\cite{Schumacher2007_switch_arxiv,CLEO_Schumacher_2008,Schumacher2007e}
on optical instability and switching analogous to the atomic case
(Sec. \ref{sec:experiment}). We concentrate on the dynamics in one
spin subsystem (say spin up) by choosing circularly polarized
excitation. We neglect a possible longitudinal-transverse (TE-TM)
cavity-mode splitting \cite{Savona1996}. As a further
simplification, we use the optical dipole selection rules and matrix
elements appropriate in quasi-normal incidence (a complete vectorial
formulation of the theory with selection rules for arbitrary angles
can be found, for example, in \cite{Schumacher2007a}). We apply a
spatial decomposition of cavity field and exciton polarization into
Fourier components $E_{\mathbf{k}}$ and $p_{\mathbf{k}}$,
respectively, with in-plane momentum $\mathbf{k}$
\cite{Savasta2003}.  The nonlinear set of coupled equations of
motion for  $E_{\mathbf{k}}$ and $p_{\mathbf{k}}$ reads
%\cite{Savasta2003,Schumacher2007a}:
\begin{align}
\label{fieldmotion}
i\hbar\dot{E}_{\mathbf{k}}=&\hbar\omega_{\mathbf{k}}^cE_\mathbf{k}
-\Omega_{\mathbf{k}}p_\mathbf{k}+i\hbar
t_cE^{\text{eff}}_{\mathbf{k},\text{inc}}\,, \\
\label{generalmotion}
i\hbar\dot{p}_{\mathbf{k}}=&\big(\varepsilon_{\mathbf{k}}^x-i\gamma \big)p_{\mathbf{k}}-\Omega_{\mathbf{k}}E_{\mathbf{k}}
\nonumber
+\sum_{\mathbf{q}\mathbf{k}'\mathbf{k}''}{\big(2\tilde{A}\Omega_{\mathbf{k}''}p_{\mathbf{q}}^\ast}p_{\mathbf{k}'}E_{\mathbf{k}''} \\
&+V_{\text{HF}}{p^\ast_{\mathbf{q}}p_{\mathbf{k}'}p_{\mathbf{k}''}\big)\delta_{\mathbf{q},\mathbf{k}'+\mathbf{k}''-\mathbf{k}}}\,.
\end{align}
The cavity-field in Eq.~(\ref{fieldmotion}) is treated in quasi-mode
approximation \cite{Savasta1996}. The effective incoming field
$E^{\text{eff}}_{\mathbf{k},\text{inc}}$ driving the field
$E_\mathbf{k}$ in the cavity mode is obtained from a simple
transfer-matrix formalism that includes the radiative width
($\Gamma=\omega\hbar^2t_c^2/(\epsilon_0cn_{\text{b}})$, with
the background refractive index $n_{\text{b}}$, the vacuum velocity
of light $c$ and dielectric constant $\epsilon_0$) of the cavity
mode and yields transmitted and reflected field components:
$E^{\text{eff}}_{\mathbf{k},\text{inc}} =
E_{\mathbf{k},\text{trans}}=E_{\mathbf{k},\text{inc}}-E_{\mathbf{k},\text{refl}}$
with $E_{\mathbf{k},\text{refl}}=-(\hbar
t_c/2n_{\text{b}}c\varepsilon_0)\dot{E}_{\mathbf{k}}$. The
cavity-to-outside coupling constant $t_c$ is chosen such that
$\Gamma\approx0.4\,\mathrm{meV}$ for $\hbar\omega=1.5\,\mathrm{eV}$.
We include excitonic PSF and HF exciton-exciton Coulomb interaction
in the nonlinear exciton dynamics in Eq.~(\ref{generalmotion});
two-exciton correlations are neglected in this study and are
expected to give merely quantitative changes because the pump is, in what follows, 
tuned far (several $\mathrm{meV}$) below the bare exciton resonance
\cite{Kwong2001b,Savasta2003} (cf. Fig.~\ref{dispersion}(a)).
Inclusion of two-exciton Coulomb correlations in our calculations
would basically lead to renormalization of $V_{\text{HF}}$ in
Eq.~(\ref{generalmotion}) and give rise to a small additional
excitation-induced dephasing \cite{Kwong2001b,Savasta2003}. The bare
exciton and cavity in-plane dispersions are denoted by
$\varepsilon^x_{\mathbf{k}}$ (with
$\varepsilon^x_{{0}}=1.497\,\mathrm{eV}$) and
$\omega^c_{\mathbf{k}}$, with
$\hbar\omega^c_{\mathbf{k}}=\varepsilon^x_{{0}}/\cos{\vartheta}$ and
$\sin{\vartheta}=|\mathbf{k}|c/(\omega n_{\text{b}})$. The dephasing
is $\gamma=0.4\,\mathrm{meV}$, $\Omega_{\mathbf{k}}=8\,\mathrm{meV}$
is the vacuum Rabi splitting, and
$\tilde{A}=A^{\text{PSF}}/\phi_{1s}^\ast(0)$. A spatial anisotropy
in the system can be modeled, e.g., by including an anisotropic
cavity dispersion $\omega_{\mathbf{k}}^c$.

In Fig.~\ref{Figswitch}, we show results where we have numerically integrated
the nonlinear coupled Eqs.~(\ref{fieldmotion}) and (\ref{generalmotion}) for
quasi steady-state pump excitation in normal incidence. The pump frequency is
tuned $5\,\mathrm{meV}$ below the bare exciton resonance. The pump (not shown)
reaches its peak intensity $I_{\text{pump}}\approx19.5\,\mathrm{kWcm^{-2}}$
shortly after $0\,\mathrm{ps}$ and is then kept constant. The total
density of excitons excited by this pump pulse is on the order of $10^{10}$
excitons per cm$^2$. We impose a slight anisotropy in the cavity dispersion
by shifting $\omega_{\mathbf{k}}^c$ to lower energies by $0.075\,\mathrm{meV}$
in directions 1 and 4. Above a certain pump threshold intensity, phase-matched
pairwise scattering of pump-induced polaritons, driven mainly by the HF term
in Eq.~(\ref{generalmotion}), leads to spontaneous (fluctuation-triggered)
off-axis signal formation (similar to \cite{Romanelli2007,verger-etal.07}).
Initially, signals in all the considered off-axis directions start to grow
simultaneously. However, as these signals grow over time, the anisotropy
(symmetry breaking) fixes the spontaneous off-axis pattern at directions 1 and
4. This can be seen in Fig.~\ref{Figswitch}(a)-(c) for times less than
$2\,\mathrm{ns}$. After $2\,\mathrm{ns}$, we apply a weak probe
($I_{\text{probe}}\approx0.1\,\mathrm{Wcm^{-2}}$) with the same frequency as
the pump frequency in direction 2 (Fig.~\ref{Figswitch}(d)). Now, the strong
off-axis emission switches to directions 2 and 5 and vanishes in the
``preferred'' directions 1 and 4. Note that the switching signal in directions
2 and 5 is about 15 times stronger than the probe pulse itself (i.e., part of
the pump is redirected from normal incidence to the directions 2 and 5). In
other words, the gain in direction 2 is $\approx11.7\,\mathrm{dB}$. When
switching off the weak probe at $\approx5\,\mathrm{ns}$, the strong off-axis
emission switches back to the preferred directions 1 and 4. The switching can
then be repeated as shown in the figure. On/off switching times in our study
are $\approx1\,\mathrm{ns}$ (corresponding to switching brought about by
$\approx13$ photons if a beam waist of $2\,\mathrm{\mu m}$ diameter is
assumed). The ns switching times imply that the bandwidth limitation (due to slow switching times) encountered in the atomic system may be significantly improved in semiconductor microcavities. However, a systematic study of the theoretical limitations of the switching times remains an important outstanding issue. In particular, a careful study of the switching times' dependence on the pump and probe power would be desirable.

Since the pump excitation is off-resonant, a relatively strong pump is
required to reach the instability threshold. In an experimental setup,
unintended off-axis scattering of pump light could reduce the contrast ratio
between ``on'' and ``off'' states and thus the performance of the switch.
However, this practical issue might be alleviated using another existing
microcavity design \cite{Diederichs2006} where resonant pump excitation could
be used. We have estimated that a reduction of the threshold intensity by two
orders of magnitude could be expected \cite{Schumacher2007e}.

A related study of all optical pattern switching has been given in Ref.
\cite{Kheradmand_2008aa}. There, a comprehensive numerical analysis of pattern
formation and switching in semiconductor microcavities is presented. It has
been found that patterns can be controlled with beams that have 100 times
smaller intensities than the intensity of the pattern, with switching times
comparable to the ones shown in Fig.~\ref{Figswitch}. The nonlinearity used in
\cite{Kheradmand_2008aa} is restricted to PSF due to the presence of
incoherent carriers, and the evaluation focuses on the positive detuning case.
As discussed above, at positive detuning strong EID from correlations may be
expected to hinder instability, although further investigations of EID from
incoherent carriers are needed to verify this hypothesis. At any rate, the
investigation in \cite{Kheradmand_2008aa} supports our belief that
semiconductor microcavities may be the most promising semiconductor system for
future demonstrations of transverse optical pattern switching at low light
levels. Experiments on stimulated polariton scattering in microcavities have
shown an impressive trend towards higher operational temperatures
\cite{Saba2001,Diederichs2006}.


In addition to microcavities, other systems may also be candidates for instabilities and switching. For example,
instabilities in the co-circular polarization channel can  be
expected in Bragg-spaced quantum wells \cite{Schumacher2007d}, which
are a specific realization of one-dimensional resonant photonic
bandgap structures), because, in these systems, a suppression of EID as
a consequence of the strong coupling between the quantum wells and
the light field (similar to the case of semiconductor microcavities
discussed above) is beneficial for optical instabilities.

We end the discussion of semiconductor systems by noting 
that, for all these systems, a substantial amount of further research is needed to experimentally verify the predicted optical switching phenomena and, once that is achieved, to
make their performance characteristics compatible with the requirements of real devices.



% section semiconductor_systems (end)

\section{Future directions} % (fold)
\label{sec:future_directions}

The initial results demonstrating optical patterns as a mechanism for all-optical switching have led to further research in both atomic vapor and semiconductor systems. There are many potential directions for future research. One immediate step forward is to extend the present numerical model described in Sec.~\ref{sec:numerical_results} to allow for pump-beam misalignment. This work is presently underway and demonstrates that the symmetry breaking introduced in this way does contribute to the overall sensitivity of the switch. Additionally, improved quantitative agreement with the experimental results from the vapor system may be obtained by using an improved numerical model that takes into account the optical-pumping nonlinearity.

Experimental verification of the switching phenomena in semiconductor systems is required to confirm the current predictions. Furthermore, such devices must then be optimized for specific applications. As this review suggests, pattern-based all-optical switches can be implemented in a variety of systems. Some, such as atomic vapor, may be ideal for ultra-low-light applications, while others, such as semiconductor systems, may be ideal for high-bandwidth applications.

% section future_directions (end)



%%% Give an acknowledgement if you have to thank someone, are
%%% supported by someone or something like that:
\begin{acknowledgement}
We thank the DARPA Defense Sciences Office Slow Light project. We thank Alex
Gaeta for providing examples of numerical models of counterpropagating beam
systems. AMCD and DJG wish to thank the Army Research Office for provided
support through grant W911NF-05-1-0228, RB, NHK and SS thank JSOP for
additional support, and ALS would like to thank ONR. SS gratefully
acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) through
grant SCHU~1980/3-1.
\end{acknowledgement}

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